Bar-Ilan University

Dynamics of Simian Immunodeficiency Virus During Acute Primary Infection in African Monkeys

David Burg

Submitted in partial fulfillment of the requirements for the Master's degree in the Faculty of Life Sciences Bar-Ilan University

Ramat Gan, Israel

2001

Acknowledgments

This research was conducted under the supervision of Dr. Avidan Neumann Faculty of Life Sciences Bar-Ilan University

Contents Abstract.....................................................................................................................................a 1. Introduction ..........................................................................................................................1 1.1 Simian Immunodeficiency Virus (SIV)......................................................................3 1.1.1 Virion Genome and Structure ....................................................................... 5 1.1.2 Viral Life Cycle............................................................................................. 7 1.1.3 Virus Entry .................................................................................................... 7 1.1.4 DNA Provirus Synthesis ............................................................................... 8 1.1.5 Nuclear Transport, Integration and Gene Expression ................................... 9 1.1.6 Virion Assembly ......................................................................................... 10 1.2 Asian monkeys .........................................................................................................10 1.3 African monkeys ......................................................................................................11 1.4 Mathematical Models in Biology .............................................................................14 2. Importance of Research ......................................................................................................16 3. Methods ..............................................................................................................................16 3.1 HIV RNA and DNA and T lymphocytes subsets measurements .............................16 3.1.1 RNA and DNA Measurements.................................................................... 16 3.1.2 Real Time Polymerase Chain Reaction....................................................... 19 3.1.3 Fluorescence-Activated Cell Sorter analysis (FACS) ................................. 19 3.2 Groups of animals ....................................................................................................21 3.2.1 African green monkeys ............................................................................... 21 3.2.2 Mandrills ..................................................................................................... 21 3.2.3 Rhesus macaques (Kaur et al.,2000) ........................................................... 21 3.3 The Model ................................................................................................................21 3.3.1 Phase plane representation .......................................................................... 24 3.3.2 Steady States ............................................................................................... 24 3.3.3 Model Stability............................................................................................ 25 3.4 Numeric integration..................................................................................................26 3.5 Fitting .......................................................................................................................27 4. Results ................................................................................................................................28 4.1 Analytical solutions ..................................................................................................28 4.1.1 Slopes and parameter estimations ............................................................... 28 4.1.2 Basic Reproductive Ratio – R0 .................................................................... 29 a

4.1.3 Approximation for Imax ................................................................................ 29 4.1.4 Estimations for other viral parameters ........................................................ 29 4.2 Kinetic Analysis .......................................................................................................31 4.2.1 Lymph node data analysis ........................................................................... 32 4.2.2 African Green monkeys .............................................................................. 32 4.2.3 Mandrills ..................................................................................................... 32 4.2.4 Macaques..................................................................................................... 33 4.2.5 Graphs and tables ........................................................................................ 33 4.3 Estimation of dynamic parameters ...........................................................................37 4.3.1 Viral doubling time (1/r0)............................................................................ 37 4.3.2 Infected-cell loss rate constant (δ)............................................................... 37 4.3.3 Target-cell infection rate constant (β') ........................................................ 37 4.3.4 Other viral parameters................................................................................. 37 4.3.5 Tables .......................................................................................................... 38 4.4 Simulations...............................................................................................................42 5. Discussion...........................................................................................................................47 6. Reference............................................................................................................................50

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Abstract Infections of naturally occurring SIV viruses in their natural hosts, African monkeys, have been shown to be nonpathogenic, while exposure of the same virus to other primates, e.g. Asian monkeys, leads to simian AIDS, comparable to HIV in humans. Mathematical modeling of viral dynamics is a powerful tool in research of viral dynamics, has changed fundamental understanding of viral behavior and has directed antiviral drug therapy. This research is the first to analyze SIV infection dynamics in its natural host, the African Green Monkeys and Mandrills. These viral-host interactions show similarities to Asian primates in virology and primary infection kinetics, but differ in their immunology and pathology Observation of SIV dynamics in African green monkeys (Cercopithecus aethiops sabaeus), mandrills (Cercopithicidae Mandrillus sphinx) and macaques (Macaca mulatta) was conducted during primary infection up to days 84, 360 and 100, respectively. Reverse transcriptase polymerase chain reaction (RT-PCR), PCR and FACS were implemented to measure viral RNA, DNA and CD4 lymphocyte concentrations in blood plasma and lymph nodes. This study defines analytical approximations of model parameters as functions of measurable viral kinetic characteristics. Our findings show that average viral doubling times were 0.85 (±0.27) days, for all animals in this study. Also, target cell infection (β ') ranged in the same order of magnitude for all animals, 4.06·10-3-2.01·10-3. Estimated viral burst-sizes (p/c) showed highly variable values, between 10-3583 virions being produced from each infected cell, unrelated to the animals' origin. Mean infected cell half-life values (t1/2) was 1.86 days (ranging from 0.46 to 6.82) in the African specimens, while the macaques showed mean values of 0.64 days (ranging from 0.07 to 1.60). Correspondingly, values of the basic reproductive ratio (R0) were at least 2.5 times higher in Asian primates, denote ing a higher infection rate in these animals. The ratio of infected cells in the entire cell pool was 100-fold higher in Asian primates. Interestingly, infected cell clearance estimations are, on average almost 3 times higher in African primate infection than pathogenic SIV infections in macaques, while other parameter estimations showed no significant differences between these two animal models. Thus, the nonpathogenic outcome of SIV infection in its natural host is identified with the loss of actively producing infected cells. We propose a qualitative distinction among chronic viral infections, a

where the pathogenicity of the viral-host interactions is dictated by this parameter. Potential antiviral defense mechanisms, targets of ongoing research, include cell dysfunction, apoptosis, genetics and cellular immune responses. These findings are important in understanding the viralhost interactions of nonprogressing HIV-positive patients, and to guide further antiviral drug investigation.

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1. Introduction Acquired Immunodeficiency Syndrome (AIDS) is a disease that has spread throughout the world. The focal points of the pandemia are areas in which the numbers of infected individuals is still increasing, especially in the poor areas of the world, in particular Africa. Currently, more than 36.1 million people carry the virus in their bodies worldwide. During the year 2000, 5.3 million people were newly infected, 70% of them in Africa. As a result of increasing awareness and advanced medical treatment, in developed countries the prevalence of the disease is decreasing. Human Immunodeficiency Virus (HIV) belongs to the family of Retroviridae, characterized by the enzyme RNA dependent DNA polymerase, better known as reverse transcriptase. This unique viral enzyme creates a DNA sequence from an RNA template and nucelosides. HIV also belongs to the group of Lentiviruses that need to integrate the proviral DNA into the host genome, utilizing the viral enzyme Integrase, in order to complete its life cycle. Another characteristic of this group of viruses is the ability to induce a chronic long-term asymptomatic infection. Advances in scientific research have found unequivocal evidence that HIV is the cause of Acquired Immunodeficiency Syndrome (AIDS). HIV illness begins with exposure to the virus and productive infection. Primary infection is characterized by the exponential increase in the number of viral particles in peripheral blood with rate r0, which attains a peak (Vmax) followed by a spontaneous decrease with rate r1 to steady state levels (Vss). Since the main target cells are CD4+ T lymphocytes, there is a decrease in the number of these cells from the initial steady state values (CD40), leading to a minimum (CD4min), and increasing to a new equilibrium (CD4ss) which is lower than the preinfection values. This chronic infection can persist without symptoms for a long period of time, of varying length between patients. There is also a high probability of transmission during viremia through sexual contact, blood and related products. After the incubation term and through an undetermined mechanism, there is an uncontrolled increase of viral replication, and destruction of the immune system at which time the patient succumbs to opportunistic infections. This stage is termed Acquired Immunodeficiency Syndrome (AIDS). The natural hosts of Simian Immunodeficiency Viruses (SIV) are African monkeys. To the best of scientific knowledge, SIV is the progenitor of the human virus HIV and the AIDS 1

epidemic. The zoontic (cross-species) infection of the human species occured before the end of the first quarter of the last century (Korber et al.,2000). The most widely used animal model in HIV/SIV research is the infection of Asian monkeys, especially macaques, with SIV. Asian monkeys are not infected in the wild, and it seems that the virus “jumped” to the macaques in captivity from sooty mangabeys (Cercobescebus atys) about 30 years ago (Hirsch et al.,1989). Similar to HIV infection in humans, infection of Asian monkeys with SIV shows the same stages: a primary acute stage, a long-term asymptomatic phase that ends with immunodeficiency and an AIDS-like illness. The validity of this model has been confirmed by genetic organization and pathogensis in vivo consistent with HIV-1 patients (Letvin and King, 1990); moreover, continuous production of virions (Müller-Trutwin et al., 1996) and a correlation between replication patterns and disease outcome (Mellors et al., 1996) show similarities to HIV1 carriers. Nowak et al. (1997) analyzed laboratory results of primary SIVsm infection in macaques, a species of Asian primates, using mathematical modeling. They calculated the logarithmic slopes of increase and decrease of plasma viral load during the first month of infection. Their results show that the viral doubling time is 6-10 hours, and that infected cell half-life was 1.3 days, on average. These results are similar to research in humans. They also investigated antiviral therapy in order to derive the down-slopes under such conditions. The data allow the calculation of the basic reproductive ratio (R0). This number estimates the mean number of target cells infected by one infected cell. R0 is important because as long as R0≥1 the infection persists, meaning that any pharmacological and/or immunology therapy must meet the condition R0<1. According to Nowak et al. (1997), any antiviral therapy needs to inhibit cell infection four-fold during primary infection in order to prevent disseminated infection, or alternatively to stop progression to AIDS disease. Similar mathematical models were also used to understand the dynamics of HIV during potent antiviral therapy (Ho et al., 1995, Perelson et al., 1996, Mittler et al., 1998). In contrast to Asian monkeys, SIV is non-pathogenic in its natural host - African monkeys. Little research has been done on infection of African green monkeys with SIV. Prevalence of infection in the wild is 45%, and there are no clinical signs associated with AIDS after infection (Ohta et al., 1988). Nevertheless, these viruses are pathogenic in certain viral-host interactions,

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since they induce AIDS-like illness in other hosts. It can be assumed that defense mechanisms help reduce the level of viremia in the natural host (Müller-Trutwin et al., 1996). Apparently, strains of SIV have co-evolved with different species of African monkeys, and therefore, each strain is dissimilar to the other. Until now all research has concentrated on the chronic phase of infection. Recently Diop et al. (2000) have carried out experiments on this model to research the immunology, virology and histology during primary infection. They show wide virus replication, which is controlled quickly and efficiently. Research of primary infection is neccesary in understanding key events in virus-host interactions. 1.1 Simian Immunodeficiency Virus (SIV) Simian immunodeficiency virus is a member of the lentiviruses, which belong to the family Retroviridae, and many of the characteristics are similar among different species and strains. The lentiviruses are exogenous, usually nononcogenic retroviruses causing persistent (chronic active) infections leading to diseases with long incubation periods. These viruses usually infect cells of the inflammatory immune system, and cause cytopathic effects such as syncytia and cell death. Lentiviral infections are not cleared by the immune system leading to accumulated damage caused by long-term immune activation over a period of many years. This important characteristic is reflected in the name of the subfamily lenti meaning “slow”. An essential trait not found in other retroviruses is their ability to infect non-dividing cells. Compared with other retroviruses, lentiviruses have a larger RNA transcript which encodes many proteins; they produce env, a large and heavily glycosylated envelope protein, and in the case of SIV a Mg+2-dependent Reverse Transcriptase (RT). Their most distinguishing property is that they encode essential regulatory and accessory genes that allow regulation of their own expression in the infected cell and signaling of the cell to the immune system. The provirus remains silent until cell activation. Replication of the lentiviruses is toxic to the cell and leads to cell dysfunction and death in infections leading to pathogenesis. The structure and replication properties of the lentiviruses are probably the reasons that the immune system is unable to eliminate the infection. Many lentiviruses affect the immune system, causing acquired immunodeficiency. Simian immunodeficiency viruses primarily infect CD4+ T lymphocytes, and the viral DNA is integrated randomly into the cellular DNA. Although many cells expressing CD4 are target cells susceptible 3

to infection, it has been shown that 90% of blood plasma virus is produced by activated productively infected CD4 T lymphocytes (Zhang et al.,1999). The number of these cells is estimated to be about 1% of the entire CD4 T cell population (Sachsenberg et al.,1998), but this estimate may be inaccurate. The exact identity or size of the target cell population cannot be ascertained by present technology. Although mechanisms of AIDS pathogenesis in humans remain unknown, much is understood about the patterns of progression from the initial HIV infection to AIDS (Coffin, 1995 and Mittler et al., 1995). During primary infection, about 3 to 6 weeks after exposure to HIV, viral titre rises to high levels, causing a brief influenza-like illness, and the density of circulating HIV decreases. Around this time the infection can be diagnosed by the presence of antibodies to HIV, known as seroconversion. The immune system is unable to entirely clear the virus and the infection enters an asymptomatic phase. During this period the CD4 T-cell population in the circulating blood gradually declines as a direct or indirect result of HIV infection (Coffin, 1995). Apparently, the immune system is capable of spending many years coping with the burden of HIV infection but after about 10 years, on average, the immune system is in total collapse leaving the host unable to combat other threatening microorganisms. At this end stage of infection viral load increases dramatically and immunodeficiency becomes clinically overt.

Figure 1a. The dynamics of HIV and CD4+ cells during HIV illness and AIDS Some scientists have suggested that HIV is not the causative agent of AIDS, arguing that explanations such as an individual's lifestyle can account for immunodeficiency (Duesberg, 1995). However, the ‘lifestyle hypothesis’ could not explain the increasing numbers of haemophiliacs, blood transfusion recipients, infants and partners of infected individuals that 4

succumbed to AIDS infection. Irrefutable scientific evidence exists proving that HIV is the infectious agent responsible for AIDS (Pantaleo et al., 1993, Coffin, 1995, Ho et al., 1995 and Wei et al., 1995). Many of the studies identifying the significance of HIV as a disease progenitor were coupled to advances in molecular biology technology (Temin and Bolognesi, 1993 and Mittler et al., 1995). 1.1.1 Virion Genome and Structure Retroviruses are enveloped, positive-strand RNA that rely on a unique enzyme, reverse transcriptase, to convert their RNA genome into a DNA provirus, and are then integrated into the cellular genome. The viral envelope is a lipid bilayer produced by the cellular plasma membrane and contains the protruding viral env glycoprotein. The composition of the core viral particle includes the p24 capsid protein and contains the viral RNA and enzymes. All retroviruses have in common the three main encoding regions: the capsid proteins (gag), the viral enzymes necessary for replication (pol), and the external glycoprotein (env) responsible for the infectivity of the viral particle. The viral enzymes encoded by pol are reverse transcriptase (RT), integrase (IN), and protease (PR). Figure 1b. shows the genomic organization of SIV.

Figure 1b. SIV Gene organization Retroviruses have one promoter and one polyadenylation site within the long terminal repeats (LTRs), and express one primary transcript. The location of the polyadenylation signal and processing site allow efficient polyadenylation only at the 3' LTR. In HIV-1, and SIV by inference, U3 and U5 region elements have been proposed responsible for polyadenylation at the 3' LTR. To produce many proteins from a single primary transcript, the retroviruses use different strategies: 1) generation and proteolytic processing of precursor polyproteins, 2) ribosomal frameshifting or suppression of translation termination, 3) alternative splicing of the primary 5

transcript and 4) bicistronic mRNAs producing two proteins. The additional proteins expressed by SIV (Table 1a) are either part of the viral particle, regulate directly viral gene expression, or interact with the cellular machinery to promote virus propagation. The additional proteins increase the complexity of the organization and expression of SIV and the other lentiviruses. Another distinguishing characteristic of these viruses is the ability to regulate their own expression via virally encoded protein factors. Some scientists propose this property to be essential for the longterm association of lentiviruses with the host and the generation of chronic active infections. Table 1a HIV/SIV proteins Encodes the capsid proteins. The precursor is the p55 myristylated protein, which is processed to gag p17 (MAtrix), p24 (CApsid), p7 (NucleoCapsid), and p6 proteins, by the viral protease. Gag associates with the plasma membrane where the virus assembly takes place.

pol

The genomic region encoding the viral enzymes protease, reverse transcriptase and integrase. These enzymes are produced as a gag-pol precursor polyprotein, which is processed by the viral protease; the gag-pol precursor is produced by ribosome frameshifting at the C-terminus of gag. viral glycoproteins produced as a precursor (gp160) and processed to the external glycoprotein

env gp120 and the transmembrane glycoprotein gp41. tat

Transactivator of HIV gene expression. Needed for HIV gene expression. Low levels of are found in persistently infected cells. Tat has been localized primarily in the nucleolus/nucleus by immunofluorescence. Extracellular tat can be found and can be taken up by cells in culture.

rev

Also necessary regulatory for HIV expression. Localized primarily in the nucleolus/nucleus, rev acts by binding to RRE and promoting the nuclear export, stabilization and utilization of the viral mRNAs containing RRE. Rev is considered the most functionally conserved regulatory protein of lentiviruses. Rev cycles rapidly between the nucleus and the cytoplasm.

vif

Viral infectivity factor, promotes the infectivity but not the production of viral particles. In the absence of vif the produced viral particles are defective, while the cell-to-cell transmission of virus is not affected significantly. Found in almost all lentiviruses. vpr detected in the cell is localized to the nucleus. It induces nuclear import of preintegration

vpr complexes, cell growth arrest, transactivation of cellular genes, and induction of cellular differentiation. It is homologous to vpx of SIVagm. Viral protein U is unique to HIV-1 and SIVcpz, however there is no similar gene in HIV-2 or SIV. Vpu is a type I integral membrane protein with at least two different biological functions: (a) vpu degradation of CD4 in the endoplasmic reticulum, and (b) enhancement of virion release from the plasma membrane. Vpu is involved in env maturation.

nef

Nef has been identified in the nucleus and found associated with the cytoskeleton. Its association with the virion is suspected but not proven. One of the first HIV proteins to be produced in infected cells, it is the most immunogenic of the accessory proteins. Initially thought to be a negative factor. The nef genes are dispensable in vitro, but are essential for efficient viral spread and disease progression in vivo. It is necessary for the maintenance of high virus loads and for the development of AIDS in macaques. Nef downregulates CD4, the primary viral receptor, and is also proposed to increase viral infectivity

vpx

Necessary for efficient replication of SIV in PBMCs. Some studies indicate that vpx and vpr proteins may be functionally distinct; yet progression to AIDS and death in SIV-infected animals infected with double mutant virus lacking both vpr and vpx was severely attenuated, whereas the single mutants were not, suggesting a redundancy in the function of vpr and vpx related to virus pathogenicity.

tev

Found primarily in the nucleus. tev contains the first exon of tat, a small part of env and the second exon of rev. It exhibits both tat and rev functions and can functionally replace both these essential regulatory proteins. It is produced very early in infection.

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1.1.2 Viral Life Cycle After the virus attaches to the cell and penetrates the plasma membrane, reverse transcriptase converts the viral RNA to DNA. The produced DNA is transported to the nucleus and integrated into the cellular DNA by the viral integrase. Due to the replication characteristics of the retroviruses, the proviral DNA is flanked by tandemly repeated sequences with important regulatory functions, termed the long terminal repeats (LTR). After integration, the retroviral provirus uses the cellular transcription machinery to express the viral RNA that has two essential roles. First, it serves as the genomic RNA that is incorporated in the virion and second, as the messenger RNA that produces all the viral proteins. The genomic RNA and the viral proteins are assembled into the viral particle, which buds out of the cell and infects new cells by attaching to specific cellular receptors. The retroviral life cycle is depicted in Figure 1c.

Figure 1c. A simplified illustration of SIV life cycle (Russell Kightley Media) 1.1.3 Virus Entry Env binds to the surface receptor CD4, which is a transmembrane glycoprotein of 58 Kd, and a member of the Ig superfamily (McDougal et al.,1986). CD4 is found on T lymphocytes, monocytes, B lymphocytes and other cells. It contributes to T cell recognition of foreign antigens in the context of major histocompatibility complex (MHC) class II determinants. CD4 interacts with MHC-II molecules and also with the T cell receptor (TCR). CD4 is the receptor for SIV, HIV-1 and also HIV-2 (Sattentau et al.,1988). Although CD4 is the primary receptor for these viruses, additional factors are necessary for infection, such as CCR5 (Easterwood, 1999). The 7

env-binding site on the CD4 glycoprotein has been mapped to amino acids 40-82. Specific activation of T cells by antigen presentation induces the CD4-TCR cointernalization. CD4 interaction with env is of paramount importance for HIV-1 and leads to infection and CD4 dysregulation. This affects the function of T cells and eventually leads to depletion of the CD4+ subset of T cells and to immunodeficiency in pathogenic infections. The importance of the envCD4 interaction for SIV and HIV is further underscored by the multiple mechanisms that lead to CD4 modulation, for example, env, vpu and nef interact with and affect CD4 at various stages. Virus entry requires the fusion of the virus and cell membrane which is mediated by the Nterminal hydrophobic region of the env transmembrane subunit (Freed et al., 1990). SIV env promotes fusion at the near neutral pH of the extracellular milieu. It is also responsible for the formation of syncytia resulting in membrane fusion among many cells. 1.1.4 DNA Provirus Synthesis After virus entry, disruption of the viral capsid occurs and the viral RT is fully activated. A ribonucleoprotein (RNP) complex forms in the cytoplasm of the infected cell and tiggers reverse transcription and transport to the nucleus. This RNP complex contains the genomic RNA together with the NC and MA protein, and the viral enzymes RT and IN. During reverse transcription, the two RNA molecules in the virion are converted to a linear double-stranded DNA. The process of reverse transcription requires priming provided by the tRNAlys3 selectively incorporated in the SIV virion, which anneales to the primer-binding site (PBS) at the 5' part of the viral genome during particle formation. The virion contains, on average, eight molecules of tRNAlys3 per two copies of genomic RNA. Elongation at the PBS results in the generation of a nascent DNA molecule of approximately 630 nt spanning the region from the PBS to the CAP site at the 5' end of the viral RNA (step 1). The Ribonuclease H (RNase H) activity specifically degrades the RNA part of the RNA-DNA hybrid of RT (step 2). This leaves the 3' end of the strong-stop DNA free to anneal to the R region of the second RNA molecule (step 3). This template 'jump' requires components of the viral capsid and depends on the affinity of RT for its template. After this, elongation by RT results in a nearly complete DNA copy terminating at the PBS, since the R and U5 regions have been removed by the RNase H (step 4). RNase H degrades the second RNA template except the polypurine tract (PPT) (step 5). Synthesis 8

of the complementary (plus-strand) DNA is initiated at the junction between the polypurine tract (PPT) and the U5 region of the LTR. Specific cleavage of the RNA-DNA hybrid at this site initiates the plus-strand DNA synthesis (step 6), which terminates within the PBS. Copying of the PBS and removal of the RNA by RNase H generates a DNA molecule that can anneal to the 3' end of the opposite DNA strand at the minus-strand PBS (step 7). This second template jump allows elongation of the plus-strand DNA to the end of a complete LTR (step 8). The completion of DNA synthesis results in a linear molecule with one complete LTR at each end (step 9). This molecule is the subject of integration after transfer to the nucleus. Side products of this process are circular molecules containing one or two LTRs, which can be detected after acute infection by HIV-1. These molecules are not able to integrate but may express proteins if transported into the nucleus.

Figure 1d. Reverse Transcription 1.1.5 Nuclear Transport, Integration and Gene Expression The double-stranded proviral DNA, complexed with proteins (preintegration complex), is transported to the nucleus of the infected cell. The integration of the proviral DNA occurs at random throughout the cellular genome by the action of viral Integrase. The integrated provirus flanked by the tandem LTRs is organized as a eukaryotic transcriptional unit. The 5' LTR contains a strong enhancer/promoter and the 3' LTR contains an 9

efficiently used polyadenylation site. Transcription of the provirus by the cellular RNA polymerase II results in a primary transcript that has two important functions: incorportation of the genomic RNA into the virion and also providing all the mRNAs encoding the viral proteins. Both viral and cellular factors regulate the viral promoter. Its activity varies greatly depending on the cell status. In many infected cells in HIV-positive individuals, virus expression remains undetectable. Thus, a state of viral latency exists in individual cells, although the infection is chronically active due to continuous expression of HIV in a fraction of the cells. 1.1.6 Virion Assembly The expressed structural proteins accumulate inside the plasma membrane. Gag also interacts with env, and Pr55gag multimerization results in the initiation of particle formation. Together with the Pr55gag, some Pr160gag-pol is also incorporated into the virion. Two molecules of genomic RNA are also encapsidated together with tRNA molecules, primarily tRNAlys3. The integration of Pr160gag-pol into the particle leads to protease dimerization and activation. The orderly cleavage of Pr160gag-pol and Pr55gag leads to particle maturation and budding. This is an essential step for production of infectious virions, since immature viral particles containing the precursor molecules are non-infectious. Cleavage also initiates the maturation and activation of the other viral enzymes. RT associates with the tRNA-viral RNA complex and initiates reverse transcription if nucleotide triphosphates are available. The accessory proteins vif and vpr, and possibly nef are also incorporated into the virion, along with cellular proteins. 1.2 Asian monkeys Asian non-human primates are not naturally infected by SIV. The first case of immunodeficiency in a non-human primate was identified in a colony of captive macaques in 1984 at the New England Regional Primate Research Center (Daniel et al., 1985). All three species of macaque (Macaca mulatta, M. nemestrina and M. arctoides) from which the virus SIVmac has been isolated are Asian monkeys. A number of lines of evidence have indicated that macaques are not naturally infected with SIV, and that they have acquired their SIV infection while in captivity. Screening of feral macaques has not revealed a single example of natural SIV infection. In addition, infection of captive macaques, although well documented, has been found to be a rare event (Desrosiers 1990). The majority of SIVmac infections have been identified in 10

retrospective studies of stored monkey tissues (Johnson et al., 1991). Thus, immunodeficiency in macaques corresponds to cross-species transmission of SIV from sooty mangabeys to uninfected macaques in captivity (Hirsch et al., 1989, Khan et al., 1991, Novembre et al., 1992). Once exposed to the virus monkeys show classic primary infection with an exponential increase in viremia, a peak and a spontaneous decrease of about 10 times in which viremia reaches a steady state. This equilibrium lasts for months and for unknown reasons plasma viral load increases concomitant with the onset of immune system failure, AIDS-like symptoms and death. Because the pattern of SIV infection in Asian monkeys is similar to HIV in humans, this animal model has been extensively studied during primary infection (Muller-Trutwin et al., 1996, Nowak et al. (1997a and others) and drug therapy (Ho et al., 1995, Nowak et al., 1997b). Little et al. (1998) have confirmed the applicability of the SIV/macaque model to understanding the HIV/human problem. 1.3 African monkeys African primates are the natural hosts of SIV. Each species has co-evolved with a speciesspecific variety of SIV (Muller-Trutwin et al.,1997). For example, African green monkeys are the natural hosts for SIVagm, while SIVcpz is the virus found in wild chimpanzes. The most prevalent example of SIV so far found in wild monkeys has been in the African green monkeys (SIVagm), with reported seroprevalence rates of 30-50% in adult animals (Ohta et al., 1988). To illustrate, sooty mangabeys (Cercocebus torquatus atys) have extremely high infection rates in North American primate centers (Fultz et al., 1986), and were found to be extensively infected in their native habitat in Africa (Marx et al., 1991). A number of other African primates have been found to harbor SIV-related virus based on serological cross-reactivity with SIV or HIV. Among feral primates known to be infected with SIV, the African green monkeys are the most numerous, most geographically dispersed and the most commonly infected (Müller et al.,1993). African green monkeys are dispersed over most of sub-Saharan Africa and have been classified as a superspecies (Cercopithecus aethiops) divided into four species: grivet (Cercopithecus aethiops aethiops), vervet (C. aethiops pygerythrus), tantalus (C. aethiops tantalus) and sabaeus (C. aethiops sabaeus) monkeys. These four species are distinguishable on the basis of phenotypic markers and have different geographic ranges: grivet monkeys reside in 11

Ethiopa and the Sudan, vervet monkeys can be found from East to South Africa, tantalus monkeys are prevalent in central Africa and sabaeus monkeys are restricted to West Africa. Interbreeding has been observed where their ranges overlap. Some authors consider the four groups as subspecies (Lernould, 1988).

Figure 1e. The geographic distribution of the four African Green monkey species in Africa. The spectrum of genetic diversity of SIVagm exceeds that of other primate lentiviruses (Johnson et al., 1989). Also, different strains of SIVagm are species-specific (Müller et al., 1993), suggesting that SIVagm viral evolution occurred with the divergence of their host species. For example, tantalus monkey viruses taken a thousand miles apart from Uganda and the Central African Republic group together in the phylogeny, whereas isolates from grivets and vervets from the same geographic location group apart. This would imply that African green monkeys have been infected with SIV for a considerable length of time, possibly even before they diverged from a common ancestor (Jin et al., 1994). This work will also focus on mandrills (Cercopithicidae Mandrillus sphinx), another African primate species known to harbor a lentivirus. Like African green monkeys, exposure of mandrills to SIVmnd, their endemic strain of SIV, causes a chronic non-pathogenic infection. They are found in a very small part of Central West Africa around Gabon, Congo and Cameroon. Until now, little research has centered on this animal, as it is an endangered species. 12

In phylogenetic studies based on gene sequences, SIVmac and SIVsm cluster closely together and in the same lineage as HIV-2 sequences (Figure 1f). Interestingly, the natural habitat of sooty mangabey coincides with the geographic pattern of HIV-2 endemicity in West Africa. It has now been generally accepted that HIV-2 infection of humans is a zoonosis, a disease communicable from animals to humans under natural conditions (Gao et al., 1992). It is not necessary to invoke vaccination programs (Hooper, 1999) or deviant sexual practices (Duesberg, 1995) to account for the transmission of SIV to humans. In many West African countries sooty mangabeys are hunted for food and kept as pets (Marx et al., 1991). Thus, humans inflicted by the scratches and bites of monkeys or the exposure to monkey blood while preparing food is quite feasible. A further line of evidence confirming the plausibility of human infection by a monkey virus was the accidental transmission of a sooty mangabey virus to a laboratory worker resulting in the establishment of a persistent infection in that individual (Khabbaz et al., 1992).

Figure 1f. Evolutionary relationships of primate lentiviruses (PLV). African primates show a similar viremia pattern during primary SIV infection to SIV/macaque and HIV (Diop et al.,2000), but this has not been found to be associated with any disease (Fultz et al., 1986 and Marx et al., 1991). Naturally occurring primate lentivirus infections 13

are not inherently non-pathogenic per se, nor are their host species intrinsically resistant to immunodeficiency, since species-specific SIV virus induces immunodeficiency in other African and Asian species. Furthermore, repeated in vivo passage of non-pathogenic viruses can generate variants with increased virulence for both natural and unnatural hosts (Fultz et al., 1989 and Hirsch and Johnson 1994). There appears to be a complex interplay between viral and host determinants at work. 1.4 Mathematical Models in Biology Mathematical modeling is a theoretical tool that helps in understanding of biological systems, and is therefore constrained by biological reasoning. One of the first population models was formulated by Verhulst in 1836 (Murray, 1993). He proposed that populations have initial exponential growth, but their numbers are limited to the enviroment, and introduced a "logistic term" into the exponential growth equation formulated by Malthus (1798). The term, 1-(N/k), decreases as the number of individuals N approaches the carrying capacity k, thereby reducing the growth rate r by an equal amount for each addition of an individual to the population. The equation: dN/dt=[r(1-N/k)]N became known as the "logistic equation" and it still serves as a way of describing the process of population growth. Volterra (1925) derived a simple model to describe the dynamics of two-species interaction systems. The model consists of two differential equations, expressing the interdependencies and the change of the populations over time. He then sought to determine the model parameters from data obtained from a fishing community in Italy. Since then, studies have made use of this model and others to interpret biological systems (Elton and Nicholson 1942 and others). As biology has developed into a quantitative science the application of mathematics becomes possible. Until the middle of the 1990’s, the principal hypothesis explaining the pattern of primary viral infection in HIV was that the immune system was playing a prominent role in controlling viral replication. Also, scientists regarded HIV to be in a state of latency during the asymptomatic chronic infection (Temin and Bolognesi 1993).

14

After inception of more sensitive and direct molecular essaying of genomic material, by polymerase chain reaction, and of cell populations, these hypotheses were tested. By using the above model, attempts have been made to give explicit estimations of viral parameters to many viruses. During 1995, two studies provided some important insights into the dynamics of HIV infection during the clinically latent period by investigating the relationship between the production of HIV-1 virions and the turnover of CD4 T-lymphocytes in patients undergoing treatment with various drugs (Ho et al., 1995 and Wei et al., 1995). These researchers found that the treatment of patients with a number of anti-HIV drugs resulted in a rapid reduction viral RNA in the plasma within a few days which was accompanied by an increase in the number of circulating CD4 T-cells. This information was then used to infer the turnover of viral particles and CD4 T-cells. It was found that approximately 109 HIV-1 virions (roughly 30% of the total) were being produced and removed from the blood of infected individuals during the chronic asymptomatic phase of HIV infection every day. Also Perelson et al. (1996) deduced virus clearance rate, infected cell half-life and viral generation time using the same model. Their results strengthened the idea that HIV is a very dynamic virus. Phillips (1996) showed that a simple mathematical model - the basic viral infection model (see section 3.3) - could explain primary infection dynamics independent of the immune response. Furthermore, Nowak et al. (1997) analyzed data from experimental infection of macaques with SIVsm, presumably a suitable animal model from a biological standpoint. They deduced the viral doubling time and infected cell half-life. Little et al. (1999) studied primary infection HIV in humans, and using analytical tools devised by Nowak et al., thereby confirming the SIV/macaque infection model applicable to the HIV in humans from a perspective of viral dynamic parameters. Neumann et al. (1998) and Nowak et al. (1997) have implemented mathematics to research the effects of antiviral therapy in SIV, HCV (Hepatitis C Virus) and HBV (Hepatitis B Virus). They also used the basic viral infection model. These investigations have shown similar trends in very different viruses. Further inquiry into the dynamics of other viruses may change our preconceptions of virology.

15

2. Importance of Research Understanding viral host interactions are paramount in designing efficient antiviral treatment. Research regarding HIV-positive nonprogressors is lacking, as there is no animal model to adequately explain the underlying reasons why some individuals seem to be immune to overt HIV- associated AIDS for long periods of time, while still harboring the virus in high titers. SIV infections in their natural hosts may be a suitable animal model for this problem. Determining the fundamental differences, which switch a viral infection to one with a pathogenic outcome or to one in which the debilitating effects of the virus are not manifest is vital, and may give insight to effectively combat chronic HIV infections.

3. Methods 3.1 HIV RNA and DNA and T lymphocyte subset measurements 3.1.1 RNA and DNA Measurements RNA and DNA measurements were carried out by polymerase chain reaction (PCR). This is a powerful technique for the amplification of trace amounts of nucleic acids, RNA or DNA. One of the most significant advantages of PCR is its extreme sensitivity, which can detect one or a few target molecules, even in non-purified crude sample preparations, due to the nature of the exponential amplification of PCR. However, this exponential nature poses the most serious obstacle to obtaining quantitative information from PCR. To perform a PCR reaction, a small quantity of the target DNA is added to a test tube with a buffered solution containing DNA polymerase, oligonucleotide primers, the four deoxynucleotide building blocks of DNA, and the cofactor MgCl2. The PCR mixture is taken through replication cycles consisting of: • the DNA is denatured into single strands for one to several minutes at 94-96ºC; • primers are allowed to anneal with their complementary sequences for one to several minutes at 50-65ºC; and • polymerase binds and extends a complementary DNA strand from each primer for one to several minutes at 72ºC.

16

• When amplifying RNA an initial step in needed, using reverse transcriptase to create the cDNA molecules to be amplified (Wang et al., 1989). Two important innovations were responsible for automating PCR. First, a heat-stable DNA polymerase, isolated from the bacterium Thermus aquaticus inhabiting hot springs. This enzyme, Taq polymerase, remains active despite repeated heating during many cycles of amplification. Second, DNA thermal cyclers were invented that use a computer to control the repetitive temperature changes required for PCR. As amplification proceeds, the DNA sequence between the primers doubles after each cycle. Theoretically, the amount of end product doubles with each amplification cycle, and the number of amplicons doubles with each cycle. However, this calculation is based on several false assumptions, the worst of which is that reaction occurs at 100% efficiency in all amplification cycles. The following factors may affect reaction efficiency: sequence of primers, sequences being amplified, their length and impurities in sample. The first three factors may affect secondary structure formation, which together with the G/C content of the target may interfere with primer binding or the melting temperature of the template. In addition, amplification efficiency may be reduced if long PCR products are produced. In addition, Q-PCR of RNA performed after a reverse transcriptase (RT) reaction is subject to other potentially confounding factors, such as the efficiency of the RT reaction and the type of primer used. Another characteristic of PCR is its plateau phenomenon: the amount of products increases exponentially only during initial cycles, then the production rate is reduced during later cycles, and eventually there is no further increase. A number of factors may be involved in causing the plateau: reassociation of PCR products competing with primer annealing and extension; low molar ratio of polymerase to target; accumulation of polymerase inhibitor (e.g. pyrophosphates); and exhaustion of other PCR components. Therefore the number of cycles needed to reach the plateau depends at least on the template sequences and on the initial concentration of template. Thus, for quantitative PCR, one addresses this efficiency problem with an internal control: a competitor amplicon. Hence, this type of reaction is known as quantitative competitive PCR (QCPCR). The internal control improves the reliability of the quantitative results by providing a

17

means to monitor and correct for the efficiency of the PCR reaction. This works best if the nature and quantity of inhibitors equally affect target and control amplicons. The ideal control amplicon is usually an artificial control, amplified with the same primer pair as the real target. It is of similar length and has a similar base pair composition to the target amplicon. However, it is important that some feature of the control distinguishes it from the target. If both amplicons are equally affected by inhibitors, use the same primers, and are of similar size and composition, they are expected to be amplified at the same efficiency. Once the control amplicon is created, a known amount of control is introduced into the sample, and amplifications run for a fixed number of cycles. After the PCR reaction is complete, the products get quantitated to determine the ratio of target to control. QC-PCR has numerous drawbacks. The main difficulty is creating and optimizing the control amplicon. However, even when this obstacle is overcome, the procedure has a limited dynamic range. Furthermore, there needs to be a method for the accurate quantitation of the end products. This can be time consuming and prevents high-throughput applications. Finally, and not uniquely to QC-PCR, contamination is always a threat. These drawbacks have severely hampered the development and use of quantitative PCR (Crotty et al., 1994). Alternatively, quantitative PCR can also be performed without internal controls by kinetic analysis (amplifications are stopped at different cycles), and the differences detected can be as small as two fold. The main disadvantage is that it generates relative differences, not absolute number of targets among samples, unless a regresion solution is employed (Wiesner et al., 1992). Many new sensitive PCR product detection systems have been developed which allow reduction of the number of cycles needed to produce detectable products before reaching a plateau. For example, automated fluorescence detection has a high ratio of signal per product molecule,

with

a

femtomolar

(10-15)

detection

level

(Sullivan

et

al.,

1992).

Electrochemiluminescence can even detect DNA at attomolar (10-18) level (Dicesare J et al., 1993), about 5 logs more sensitive than the ethidium bromide staining detection (Sarkar et al., 1992).

18

3.1.2 Real Time Polymerase Chain Reaction Real-time PCR quantitates the initial amount of the template most specifically, sensitively and reproducibly, and is a preferable alternative to other forms of quantitative PCR which detect the amount of final amplified product. Real-time PCR monitors the fluorescence emitted during the reaction as an indicator of amplicon production during each PCR cycle in real time, as opposed to the endpoint detection by conventional quantitative PCR methods. The real-time PCR does not detect the size of the amplicon and thus does not allow the differentiation between DNA and cDNA amplification. However, it is not influenced by non-specific amplification. Real-time PCR quantitation eliminates post-PCR processing of PCR products, which is necessary in competitive PCR. This helps to increase throughput, reduce the chances of contamination and remove post-PCR processing as a potential source of error. In comparison to conventional PCR, real-time PCR also offers a much wider dynamic range of up to 107-fold compared to 103-fold in conventional PCR. This means that a wide range of ratios of target and normalizer can be assayed with equal sensitivity and specificity. The real-time PCR system is based on the detection and quantitation of a fluorescent reporter. This signal increases in direct proportion to the amount of PCR product in a reaction. By recording the amount of fluorescence emission at each cycle, it is possible to monitor the PCR reaction during the exponential phase, where the first significant increase in the amount of PCR product correlates to the initial amount of target template (Souaze et al., 1996). 3.1.3 Fluorescence-Activated Cell Sorter analysis (FACS) CD4 counts were determined by fluorescence-cctivated cell sorter analysis using the FACScan (Becton-Dickson, Heidlberg, Germany). Cells being analyzed by FACS may be alive or fixed at the time of measurement, but must be in monodisperse suspension. They are passed single-file through a laser beam by continuous flow in a fine stream. Each cell scatters some of the laser light, and emits fluorescent light excited by the laser. The cytometer typically measures several parameters simultaneously for each cell: • low angle forward scatter intensity, approximately proportional to cell diameter • orthogonal (90º) scatter intensity, approximately proportional to the quantity of granular structures within the cell 19

• fluorescence intensities at several wavelengths Light scatter is commonly used to exclude dead cells, cell aggregates, and cell debris from the fluorescence data. It is sufficient to distinguish lymphocytes from monocytes from granulocytes in blood leukocyte samples. It is also used to quantitate aggregation of living cells. Fluorescence intensities are typically measured at several different wavelengths simultaneously for each cell. By making surface receptors fluorescent, they can be measured. Fluorescent probes are used to report the quantities of specific components of the cells. Fluorescent antibodies are often used to report the densities of specific surface receptors, and thus to distinguish subpopulations of differentiated cell types. Intracellular components for which an antibody is available can also be reported by fluorescent probes. Flow cytometry can also monitor rapid changes in intracellular free calcium, membrane potential, pH or free fatty acids. Flow cytometers involve sophisticated fluidics, laser optics, electronic detectors, analog-todigital converters and computers. The optics deliver laser light focused to a beam a few cell diameters across. The fluidics hydrodynamically focus the cell stream to within a small fraction of a cell diameter. The electronics quantitate the faint flashes of scattered and fluorescent light. The computer records data for thousands of cells per sample, statistically analyzes and displays the data graphically. Unlike cell sorters, these instruments cannot separate cells into different containers based on their properties; the samples are consumed and discarded during analysis. It is a closed fluidic system, so analysis of biohazardous samples (such as human blood samples) is possible with appropriate precautions and authorization. The FACScan uses a 15mW output air-cooled argon gas laser, with a fixed wavelength emission of 488nm. It has three fluorescence detection channels, which simultaneously detect green, yellow-orange, and red light. Fluorescein is used extensively for the green channel, and phycoerythrin or propidium iodide, for staining DNA, for the yellow-orange channel. Dyes are also available which can be excited at 488 nm that emit in the red. It can analyze cell suspensions at the rate of several hundred cells per second. Data are saved to the hard disk, where they can later be analyzed with graphics software.

20

3.2 Groups of animals 3.2.1 African green monkeys In cooperation with Dr. Michaela Müller (Pastuer Inst., Paris), we have researched primary infection of the virus SIVagm.sab in African green monkeys. Blood samples were taken from four animals twice a week during the first three weeks (on days 3,7,10,14,17,21) and on days 28, 35 and 84, postinfection; and from three animals taken twice a week for three weeks postinfection. Viral load was quantified by three distinct limiting dilution RT-PCR tests in duplicate or quadruplicate and real time PCR was used to validate the data. The limit of detection was measured at 100 copies RNA·ml-1. T lymphocyte CD4+/CD3+ and CD8+/CD3+ levels were measured by fluorescent staining and fluorescence-activated cell sorter (FACS) analysis and normalized to white blood cell (WBC) counts the same day. Also lymph node biopsies were collected once a week and were subjected to limiting dilution PCR. 3.2.2 Mandrills Primary infection data in mandrills, another species of African monkeys, exposed to SIVmnd-1 was also received from Dr. Müller’s group. Blood samples were taken twice weekly until day 21 and on days 28, 32, 60, 180 and 360 postinfection. A limiting-dilution RT-PCR assay specific for SIVmnd-1 was employed to quantify viral RNA in blood plasma, and its detection limit ascertained at 210 copies RNA·ml-1. CD4 counts were attained by FACS. 3.2.3 Rhesus macaques (Kaur et al.,2000) For comparison we received plasma viral load and lymphocyte data from rhesus macaques infected with SIVmac251 or SIVmac239 (Kaur et al., 2000). Samples were collected every week and analyzed by QC-PCR and FACS respectively. 3.3 The Model The basic model used in the research of viral dynamics is one that describes the interactions among target cell, infected cell and virus populations. This model has been the focus of biomathematical studies of viral infection in vivo. Influx of target cells is assumed to be from a constant source, since the thymus and secondary lymphoid tissue begin to collapse many weeks after infection and for lack of other information regarding target cell dynamics. Target cell death 21

is assumed to proportionate to the number of cells. Virus infects target cells, and infected cells are lost at a constant rate. Infected cells produce and release virions to the extracellular region, and virus is cleared from the body. Figure 3a shows a graphical depiction of this model.

s Target cell population

Infected cell population

d

δ β p

Viral load c

Figure 3a: Basic biological model of viral infection dynamics The basic model can be translated into a system of differential equations, that describes the interaction that occurs among populations of target cells (T), infected cells (I) and free virions (V).

dT = s − dT − βVT dt

(1)

dI = βVT − δI dt

(2)

dV = pI − cV dt

(3)

Equation 1 expresses the kinetics of target cells where T is the concentration of target cells, s and d are the production rate and rate of loss constants of target cells. β is the infection rate constant of target cells by virus, therefore βVT is the number target cells that become infected cells; and is also the chance of a productive collision between a virion and a target cell. Equation 2 expresses the kinetics of infected cells where I is the concentration of infected cells, δ is the loss rate of infected cells. Equation 3 expresses the dynamics of the virus where V is the concentration of virions, p is the production rate constant of virions by infected cells and c is the virus clearance rate constant. Viral dynamics have been shown to be very rapid for SIV by Müller-Trutwin et al. (1996), for HIV by Ho et al. (1995) and Perelson et al. 1996, and for HCV by Neumann et al. (1998). 22

Consequently, V is constrained only by the growth of I and a quasi-steady state can be assumed: dV/dt=0; therefore V≈pI/c. This assumption is sufficient for ratios of p/c that are large, denoting a rapid viral turnover rate. Also for the first few days we assume that target cells remain more or less constant in their preinfection steady state T0=s/d, as described by Akari et al. (1999). In this way we derive a simplified model for viral infections:

 dT  dt = d (T0 − T ) − β ' IT p , where β ' = β  c  dI = β ' IT − δI  dt

(4)

where d is the target-cell death, δ is the infected-cell loss and β’ are the target-cell infection rate constants. As implied by the quasi-steady state, virus is a linear function of infected cells. This model must conform to biological rationale, meaning that values of parameters and of populations are positive (>0). In the absence of clinical data about viral dynamics, this model assumes constant rates for all parameters. Figure 3b. depicts the kinetics of system (4) using parameters assumed by Phillips (1996).

4 logV

5

4 3 3

2

log[target cells]

log[viral load]

logT

2 0

50 100 150 Time (days postinfection)

200

Figure 3b. Numeric integration result of equation (4) Plasma viral load increases exponentially to a peak value, and decreases spontaneously to a steady state level through dampening oscillations. Target cell concentration remains unchanged until the viral load reaches a threshold state, then declining to a minimum and oscillating to a steady state level lower than the initial steady state. Generally, primary viral infection data do not show oscillatory behavior. This behavior is similar to what is seen in the quantitative datasets for HIV. 23

3.3.1 Phase plane representation Phase plane representations are the graph of one variable as a function of another. In this case, viral load as a function of target cell availability. These graphs give insight into the important stages of viral infection. Initially, the virus increases while target cells stay almost constant. As target-cell concentration decreases and viral load grows past its steady state, viral growth rate also decreases. Peak viral load coincides with the target cell steady state. Target cells continue to decline rapidly to a minimum corresponding with the most rapid reduction of virus in blood plasma.

log[viral load]

V max

5 V

4

3 2.4

T min

T

2.7 log[target cells]

T0

3

Figure 3c. Phase plane graph of viral infection 3.3.2 Steady States Steady states of the model are conditions where there is no change in T, I and V. Definition of the equilbria of the system in mathematical terms is when dT/dt=0, dI/dt=0 and dV/dt=0. Because system (4) is modeling a biological system, this study is interested only in non-negative equilibria. Formalization of the steady states as functions of viral parameters gives the preinfection and postinfection equilbrium values, e1(s/d,0) and e2(δ/β’,s/δ-d/β’) respectively. These solutions are correct on condition that R0=sβ’/dδ>1. Equilibrium for the differential equation for V is given using the quasi-steady state.

24

3.3.3 Model Stability Analysis of the model gives important information about the behavior of the system. Designating steady states and their values is paramount in solving the system of equations, but the stable nature of the entire system or of each steady state is unknown. A general look at stability can be shown through nullclines. The definition of a nullcline is when the derivative is equal to zero. Plotting the nullcline functions on a phase plane gives information on the vectors around the steady states (Murray 1993). If the vectors converge on the steady state then it is stable, but the equilbria are unstable if the vectors diverge from it. This is a geometric method for finding the equilibria and extending our understanding of the solutions of the differential equations. 30 25

Infected cells

20 15 10 5 0 -5

200

700

1200

Target cells

Figure 3d: Nullcline representation of system (4) In the case of nonlinear systems, we only know how to describe the solutions globally, via nullclines. The Jacobian matrix analysis approximates a nonlinear system by a linear one around the equilibrium point. The assumption is that the behavior of the solutions of the linear system approximate the nonlinear one. The Jacobian matrix yields the general vectors around the steady states and this determines the specific stability of each steady state. At the preinfection steady state system (4) is at equilibrium e1 where T=s/d and I=0. Therefore:

 − d J e1 =   0 

−

β 's 

d   β 's −δ  d 

Postinfection equilibrium e2 is characterized by t → ∞ , meaning T=δ/β’ and I=s/δ-d/β’therefore:

 β 's  − δ J e2 =  β 's  −d  δ 25

 −δ   0  

The determinant and trace of the matrix  a 11 

 a 21

a 12   are defined by a11·a22-a12·a21 and a11+a22, a 22 

respectively. The steady state will be stable when DetJ>0 and TraceJ<0 (Keshet, 1993). Both matrices show that in order to have a biologically reasonable stable steady state, the criterion of s/d<δ/β’ must hold. Meaning that the initial value of T must be larger than its steady state value. These analyses indicate that e1 is unstable of the “saddle” variety, and that e2 is a stable focus. The eigenvalues of the Jacobian matrix characterize the fate of the solutions around the equilibrium point from the eigenvalues. Negative or complex eigenvalues with a negative real part, then the equilibrium point is a sink .If the eigenvalues are positive or complex with positive real part, then the equilibrium point is a source (that is all the solutions will move away from the equilibrium point. If they are complex, then the solutions will spiral around the equilibrium point. (for the sink) or will spiral away (for the source). When the eigenvalue is a real number with different sign (one positive and one negative), then the the equilibrium point is a saddle. In fact, there will be two solutions which approach the equilibrium point, and two more solutions which approach the equilibrium point. The eigenvalues are derived from: λ2+(TraceJ)λ+DetJ=0 Eigenvalues will have imaginary components when ∆=(DetJ)2-4TraceJ<0 (Perko, 1991). Solution of the eigenvalues of system (4) as functions of parameters shows that d and δ assert the most influence on the oscillatory behavior of the system (4). Oscillations are very conspicuous when δ»d, but as δ approaches d the oscillations become less prominent (Zhang and Neumann, 2001). 3.4 Numeric integration Many differential equation systems cannot be solved analytically, including the basic model of viral infection. In order to achieve an approximate solution this study utilizes the computerassissted simulation program "Madonna" (Oster and Macey, 2000). Programs such as these perform numerical integration, i.e., they use initial conditions and calculate the numerical solution for each equation, repeating the calculation every unit of time. The solution is then displayed graphically on screen. These graphs follow the kinetics of the system as a function of parameters, and it is possible to examine the influence that modification of parameters have on the behavior of 26

the system. It is also possible to examine the effects of assumptions and approximations that can facilitate and confirm the analytical solutions of the system. 3.5 Fitting After confirming that the kinetics of the model are similar to empiric findings and estimating viral parameters, the model is fitted to each set of data in order to acquire optimal values for each of the parameters in the model, as well as determining elusive parameters. Fitting is accomplished in three ways: 1. Manual fitting by applying analytical parameter solutions and using an electronic spreadsheet, such as Microsoft Excel (Microsoft 1997). 2. Non-linear fitting using a computer program such as Madonna (Oster and Macey 2000), which fits parameter values to the model by searching multidimensional space for the maximal gradient.

27

4. Results 4.1 Analytical solutions Deriving analytical solutions of the model’s differential equations using, approximations based on biological assumptions, allow mathematical formulation of model parameters. Analytical solutions for the exponential growth, peak and decline of viremia are tools in understanding the relationships in the biological system. These tools also aid in fitting the model to clinical data. 4.1.1 Slopes and parameter estimations A prediction of viral dynamics shows that I increases rapidly while T remains more or less constant during the first days after infection (see figure 3b). By setting T=T0 in the infected cell equation (2), we derive an expression for the exponential up slope (r0). Solution of

dI = ( β 'T0 − δ ) I is: dt I = I 0e β 'T0 −δ and the slope of this function is:

r0 = β 'T0 − δ

(5)

Deriving the maximum down slope (r1) is similar to r0, and since it coincides with Tmin (see figure 3b):

r1 = − β 'Tmin + δ (Nowak et al 1997b). Because of difficulties in defining T0 and Tmin as described above (section 1.1.1), we designate the ratio between them as:

k=

Tmin . Substituting k into r1 we obtain: T0 r1 = −kβ 'T0 + δ

(6)

Indeed these solutions are maximal estimates of parameters measurable from kinetic data. Because the equations for r0 and r1 are a two equation system with two variables, δ and β’, the solution is immediate, and we can derive the infected cell loss rate constant (δ):

δ =

r0 + r1 − r0 1− k 28

(7)

We can also determine the target-cell infection rate constant (β’):

β '=

r0 + r1 1 ⋅ 1 − k T0

(8)

δ is a function of the primary infection slopes, but β’ is also a function of T0. As state above the exact determination of T0 is impossible. Because initial slopes values are maximal estimates, δ and β’ are minimal estimates of the infected cell loss and target cell infection rate constants. These approximations are useful but are dependent on T0. Therefore one more analytical solution will allow an independent estimation. 4.1.2 Basic Reproductive Ratio – R0 The basic reproductive ratio is the average number of secondary infections arising from one infected cell during its life span. Nowak et al. (1997) defines this number as the dominant root of:

r0 + r0 (δ + c) δc 2

R0 = 1 + When c » r0+δ, R0 → 1 +

r0

δ

; and by substituting the solution for δ (7) we get:

R0 = 1 +

r0 (1 − k ) kr0 + r1

(9)

4.1.3 Approximation for Imax By solving the trajectories, determination of the maximum value of I becomes possible. Solving dI/dT derives the trajectories of this model. Unfortunately the model is too complex, and this mathematical operation is unintegrable. However assuming − β ' IT » d (T0 − T ) , our model becomes the epidemiological model SIR (Murray 1993), and the trajectories solution is straightforward.

dI (β 'T − δ )I ≈ δ − 1 = dT d (T0 − T ) − β ' IT β ' T T δ T1 ∫ dI = β ' T∫ T dT − T∫ dT I I

0

I (T ) =

0

0

δ T ln − T + T0 + I 0 β ' T0

Since Imax coincides with the postinfection target-cell steady state ( T = δ β ' , see figure 3c),

δ  δ δ δ I   = ln − + T0 + I 0  β '  β ' β ' T0 β ' 29

Initial values of infected cells (I0) are assumed to be very small, about 10-5 cells·µl-1 (Stafford et al., 2000), therefore:

I max ≈ T0 −

δ (1 + ln R0 ) ; R0 = β 'T0 β' δ

We can now substitute R0 with equation (9), which is a function of the initial up slope and the infected-cell loss rate constant. By implementing this expression for R0 we derive an approximation for Imax as a function of preinfection target cell value and initial infection slopes.

I max = T0 −

δ (1 + ln R0 ) β'

(10)

Since Imax is derived from a model, which does not include target cell population dynamics, it represents a minimal approximation. Actually, the assumption − β ' IT » d (T0 − T ) is the specific example of the model where d → 0. If d>0 then target-cell population dynamics increase and this estimation will underestimate Imax by an indeterminable amount. Even so, values of d that retain biological meaning stay small, generally 0
180 160 d=0 d=0.01 d=0.05 d=0.1 Imax

Infected cells (cells/ l)

140 120 100 80 60 40 20 0 0

200

400 600 800 Target cells (cells/µl)

1000

Figure 4a. Trajectories of the viral infection model, assuming 0
30

4.1.4 Estimations for other viral parameters Now estimation of the viral burst size may be accomplished from the quasi-steady state assumption, V ≈ pI/c:

p = c

Vmax  kr + r  T0 1 − 0 1 (1 + ln R0 )  r0 + r1  

(11)

where Vmax is the maximal viremia measured during primary infection. Equation (11) shows the dependence of the viral burst size on preinfection target cell value. That means, any change in T0 will be compensated for by p/c. By applying the quasi-steady state, substituting analytic approximations and rearranging we can determine the viral infection rate constant of target-cells (β):

βVmax = β ' I max β =

, β’=βp/c

β 'T0 − δ (1 + ln R0 )

β=

Vmax r0 − δ ln R0 Vmax

(12)

which is a function of the initial slopes, maximal viremia (Vmax) and the Tmin/T0 ratio, but unlike β’ is independent of the absolute value of T0. Determination of the target-cell intrinsic death rate constant d is straightforward by using the above parameters and the V steady state solution:

V =

s p d ⋅ − , s ≈ dT0 δ c β

T p 1  d =V  0 ⋅ −  δ c β  And s is given by:

s = dT0

−1

(13) (14)

4.2 Kinetic Analysis For each AGM, maximum up (r0) and down (r1) logarithmic slopes were calculated using mean viral load data from days 3-7 and 14-17 postinfection respectively. Also lymph node viral RNA exponential slopes were calculated. For the second AGM group, exponential viremia slopes were estimated between days 5-7 and 14-17, and mandrill’s slopes were calculated for days 4-7 and 14-17, postinfection. Macaque primary infection slopes were calculated for days 7-14 and 1421 or 28 as data allowed. Also maximum viremia, CD4+ subset preinfection and minimal values 31

were noted for all animals. Steady state values for virus and lymphocytes were determined by calculating the mean of long term infection. Duplicate data of the first group of African green monkeys was statistically analyzed to ensure uniformity of the data. 4.2.1 Lymph node data analysis HIV/SIV infection occurs in lymph nodes. Inspection of lymph node tissue viral kinetics in four African green monkeys established that compartmental and mixed infection are comparable. In fact, lymph node and blood plasma viral kinetics showed similar dynamics (see figure 4b); albeit, up slopes in lymph nodes were 31-62% higher than in blood plasma, and down slope values showed a marked increase of 2.5-3 times (see table 4a). Hence, the assumption that blood viremia is a reflection of the status at lymph node level holds. 4.2.2 African Green monkeys African green monkeys infected with SIVagm show classical primary infection dynamics exponential increase in viremia, reaching a peak and receding ~2log to steady state levels (see figure 4b). Initial logarithmic up slopes ranged from 0.75-3.48 (1.78±0.91 on average); and down slopes ranged 0.46-1.46 (average of 0.91±0.40). Also, mean peak viremia was 5.13·107 copies RNA·ml-1, and mean viral steady state was 1.62·105 copies RNA·ml-1 ~100-fold lower than peak values. Average initial value of CD4 lymphocytes was 1,464±842 cells·µl-1 (see table 4c). Duplicate viral data was statistically analyzed, and the average variability was 2.87%. The logarithmic slopes were also compared to check the consistency of the data; the mean variability was also acceptable, 2.99%. 4.2.3 Mandrills Mandrills showed kinetics very similar to African green monkeys, even though CD4 lymphocyte dynamics were not as striking (see figure 4d). Average up slopes and down slopes were 1.34±0.54 and 0.95±0.35, respectively. Mean peak viremia was 6.92·107 copies RNA·ml-1; and viral steady state values were also ~100-fold lower, 1.86·105 copies RNA·ml-1 on average. Initial CD4 lymphocytes values were lower than noted by earlier observations, around 696±122 cells·µl-1 (see table 4c).

32

4.2.4 Macaques Macaques also demonstrated classical primary infection after SIV infection (see figure 4d). The average up slope value was 1.12±0.27, but down slopes were significantly slower than African primates, 0.15±0.08 on average. Peak viremia averaged 5.25·107 copies RNA·ml-1, while steady state viral levels were only slightly lower at 1.45·107 copies RNA·ml-1. Initial CD4 lymphoctye values were within normal range, 1,708±630 cells·µl-1. 4.2.5 Graphs and tables

3

2

1

0 0

10

20 30 40 days postinfection

7 plasma lymph node

6

6

7 6

5

5

4

4

3

3

2

2

1 0

50

10

20 30 40 days postinfection

50

Figure 4b. African green monkeys - lymph node and blood plasma viral kinetics (mean of all 4 animals)

Table 4a. African green monkeys - lymph node and blood plasma viral kinetics

Lymph node 96001 96008 96011 96023 Mean ±

-0.98 0.77 1.15 0.97 0.19

0.99 0.99 1.73 1.15 1.21 0.43

Up slope Down slope

0.77 1.15 1.15 0.58 0.91 0.29

Viral load

Infected PBMC

Up slope Down slope Plasma 96001 96008 96011 96023 Mean ±

mean log[copies RNA/10 LNC]

mean log[copies RNA/ml]

4

6

mean log[copies DNA/10 PCMB]

8 plasma lymph node

0.33 0.66 0.33 -0.44 0.19 33

Plasma 96001 96008 96011 96023 Mean ±

1.41 -1.26 0.90 1.19 0.31

-1.46 1.34 0.90 1.23 0.46

Lymph node 96001 96008 96011 96023 Mean ±

1.24 1.57 1.60 1.60 1.50 0.17

-0.33 0.66 0.33 0.44 0.19

4 VL CD4

7

mean log[cells/µl]

mean log[copies RNA/ml]

8

6

3

5 4 3 2

2 -20 -10

0

10

20

30 40 50 60 Days postinfection

70

80

90 100

Figure 4c. African green monkeys - mean viral load and CD4 cell concentration

Table 4b. African green monkeys - kinetic parameter analysis CD40 CD4ss CD4min

log[Vmax]

-1 cells·µl-1 cells·µl-1 cells·µl-1 log[copeis RNA·ml ]

day at Vmax

log[Vss]

day

log[copeis RNA·ml-1]

VL up VL down slope r0 slope r1

96001 96008 96011 96023 98007 98008 98011

932 487 1022 1195 1552 3015 2045

654 387 682 626 982 1273 1018

283 246 314 550 621 538 507

7.86 7.32 8.23 7.11 7.55 7.32 7.39

10 10 12 10 7 8 9

5.47 5.45 5.75 5.55 4.69 5.35 5.58

1.41 0.75 1.26 0.90 3.48 1.61 2.04

0.46 1.46 1.34 0.90 0.51 0.46 0.69

Mean ±

1464 842

803 300

437 151

7.71 0.53

9.4 2.1

5.21 0.44

1.78 0.91

0.91 0.40

34

4 8 log[copies RNA/ml]

6 3

5

log[cells/µl]

VL CD4

7

4 3 2

-30 -20 -10

2 0

10

20

30 40

50

60

70

80

90 100

Days postinfection Figure 4d. Mandrills - mean viral load and CD4 cell concentration

Table 4c. Mandrills - kinetic parameter analysis CD40 CD4ss CD4min cells·µl

-1

cells·µl

-1

cells·µl

-1

day at Vmax

log[Vss]

log[copeis RNA·ml ]

day

log[copeis RNA·ml-1]

log[Vmax] -1

VL up VL down slope r0 slope r1

10G 12A4 12C2 2C2

794 782 675 531

938 697 551 338

492 708 518 273

7.37 8.17 7.67 8.17

7 7 12 13

5.27 5.27 5.27 5.27

1.07 1.07 1.07 2.15

0.58 1.38 0.77 1.07

Mean ±

696 122

631 252

498 178

7.84 0.39

9.8 3.2

5.27 --

1.34 0.54

0.95 0.35

35

4 mean log[copies RNA/ml]

7 6 3

5 4

mean log[cells/µl]

VL CD4

8

3 2

2 0

10

20

30

40

50

60

70

80

90

100

Days postinfection Figure 4e. Macaques - mean viral load and CD4 cell concentration

Table 4d. Macaques - kinetic parameter analysis CD40

CD4ss

log[Vmax]

day at Vmax

log[Vss]

day

log[copeis RNA·ml-1]

-1 cells·µl-1 cells·µl-1 log[copeis RNA·ml ]

VL up VL down slope r0 slope r1

256 260 285 303 409 410 239 249 265 325

1355 1949 3157 1398 2085 1578 1573 1879 796 1313

332 887 753 581 1171 1407 1573 2069 506 1316

7.45 7.49 8.45 8.00 8.45 7.96 7.04 7.85 6.28 8.26

14 13 14 7 14 14 14 14 7 10

6.21 7.31 7.38 6.59 7.87 7.41 7.23 8.09 6.60 6.95

1.17 1.33 0.90 1.62 0.90 0.82 0.99 1.37 1.26 0.86

0.20 0.07 0.19 0.22 0.20 0.17 0.08 0.03 0.05 0.24

Mean ±

1708 630

1060 545

7.72 0.68

12.1 3.0

7.16 0.59

1.12 0.27

0.15 0.08

36

4.3 Estimation of dynamic parameters 4.3.1 Viral doubling time (1/r0) The viral doubling time is a function of the primary up slope of viral load. Table 4e shows the minimal estimates of viral doubling time during primary infection for all specimens. All values are in the same range, 0.79-0.95. 4.3.2 Infected-cell loss rate constant (δ) Minimum and maximum evaluations of infected cell loss, half-life and number of secondary infections are shown in table 4f. African green monkeys and mandrills infected with SIV show much higher values of infected cell loss rate constant compared with macaques, values estimated for macaques range from 0.07 to 0.29, for African green mokeys 0.54 to 1.57 and for mandrills 0.58 to 1.62. The basic reproductive number was 1.25-2.83 times higher in macaques than in their African counterparts. These results show values of δ that are significantly lower (p<0.005), in Asian primate infection than chronic SIV infection in its natural host. 4.3.3 Target-cell infection rate constant (β') The target cell infection rate constant, assuming cell-to-cell viral transmission, was similar between African and Asian primates. Calculated maximum and minimum values ranged between 0.7-3.8·10-3 (see table 4g). Higher values for mandrills (3.2-9.7·10-3) are attributable to the small ratio between preinfection and minimal CD4 counts. 4.3.4 Other viral parameters Table 4h shows our minimum and maximum approximations of viral infection rate of target cells, viral burst-sizes, loss rate constant of infected cells and their proportion out of the entire cell pool. The viral infection rate of target cells (β) was equivalent in all animals tested, and average values for mandrills, macaques and African green monkeys were in the same order of magnitude: 9.3·10-6, 1.5·10-5 and 2.2·10-5, respectively. Additionally, all groups showed comparable average burst-sizes (p/c), from 131 to 893 copies of RNA per cell. These are not unique solutions as the absolute number of initial target cells is not known. Even so, any change in T0 can be compensated for by the viral burst-size constant (see equation (11)). In any event the above calculations assumed a maximal limit for T0. 37

Hence the calculated value of p/c is a minimal approximation. We also calculated the fraction of infected cells actively producing virulent virions out of the entire cell pool, during peak viremia. The macaques show a high percentage of target cells being infected 29-58%. Yet African primates exhibited a lower portion of infected cells, 3-31%. Inference of the intrinsic target cell death rate constant (d) using equation (13) yielded values causing exceptionally oscillatory behavior in the simulations. Therfore this parameter was determined by fitting this parameter to each dataset. 4.3.5 Tables Table 4e. Viral doubling time Viral t2 days

AGMs 96001 96008 96011 96023 98007 98008 98011

0.71 1.33 0.79 1.11 0.50 0.62 0.49

Mean ±

0.79 0.22

Mandrills 10G 12A4 12C2 2C2

0.93 0.93 0.93 0.47

Mean

0.82 0.22

± Macaques 256 260 285 303 409 410

0.86 0.75 1.12 0.62 1.12 1.22

Mean

0.95 0.22

± 38

Table 4f. Infected cell loss rate, half-life and average number of secondary infections

δ

Infected cell half-life -1

day

R0

days

#

max k

min k

min k

max k

min k

max k

96001 96008 96011 96023 98007 98008 98011

1.27 3.71 2.49 2.44 2.25 1.01 1.59

0.46 1.46 1.34 0.90 0.55 0.54 0.69

0.79 0.27 0.40 0.41 0.44 0.99 0.63

2.19 0.69 0.75 1.11 1.82 1.84 1.44

2.11 1.20 1.51 1.37 1.89 2.59 2.28

4.10 1.52 1.94 2.00 4.63 3.96 3.95

Mean ±

2.11 0.92

0.85 0.40

0.56 0.25

1.40 0.58

1.85 0.51

3.16 1.28

12A4 10G 12C2 2C2

-3.27 6.82 4.47

1.38 0.58 0.77 1.07

-0.31 0.15 0.22

0.72 1.73 1.30 0.93

1.04 1.33 1.16 1.48

1.78 2.86 2.40 3.00

Mean

4.85 1.81

0.95 0.35

0.23 0.08

1.17 0.44

1.25 0.19

2.51 0.55

256 260 285 303 409

0.65 1.24 0.53 1.52 1.60

0.20 0.07 0.19 0.22 0.20

1.54 0.80 1.89 0.66 0.62

4.88 13.58 5.30 4.63 4.98

2.80 2.07 2.70 2.07 1.56

6.69 19.06 5.75 8.51 5.46

Mean ±

1.11 0.50

0.18 0.06

1.10 0.58

6.68 3.87

2.24 0.51

9.09 5.70

66

Mandrills

± Macaques

39

Table 4g. Target cell infection rate constant

β' (·10-3) µl·cells-1·day-1 max k min k

AGMs 96001 96008 96011 96023 98007 98008 98011

2.88 9.18 3.67 2.80 2.74 0.87 1.78

2.01 4.54 2.54 1.51 1.64 0.71 1.34

Mean ±

3.42 2.69

2.04 1.24

12A4 10G 12C2 2C2

-5.47 11.69 12.45

3.14 2.08 2.73 6.06

Mean ±

9.87 3.83

3.50 1.53

256 260 285 303 409

1.34 1.32 0.45 2.25 1.20

1.01 0.72 0.34 1.31 0.53

Mean ±

1.31 0.64

0.78 0.39

Mandrills

Macaques

40

Table 4h. Viral infection and loss rates of target cells, burst size and I:T ratio

β (·10-6)

p/c

d -1

I/(I+T) at Imax -1

µl·copies RNA ·day max k min k

max k

min k

max k

min k

max k

min k

96001 96008 96011 96023 98007 98008 98011

5.01 3.21 1.41 10.44 16.75 30.91 29.49

8.32 6.96 2.18 21.49 34.05 41.17 44.20

575 2859 2606 268 163 28 60

241 653 1167 70 48 17 30

1.34 4.41 1.56 5.12 0.60 1.00 4.71

0.80 3.75 1.29 3.90 0.30 0.72 3.07

0.32 1.33 0.33 1.41 0.09 0.25 0.83

19.8 1.87 8.77 7.11 17.5 36.9 28.7

Mean ±

13.89 12.26

22.62 17.36

937 1242

318 437

2.68 1.97

1.97 1.54

0.65 0.54

17.2 12.5

12A4 10G 12C2 2C2

0.15 6.19 1.63 2.69

1.91 19.92 8.56 6.60

-883 7166 4636

-104 319 919

0.67 3.54 1.94 1.05

0.46 2.01 1.15 0.62

0.13 0.80 0.40 0.13

0.10 2.75 1.17 8.56

Mean ±

2.67 2.57

9.25 7.64

4228 3161

747 692

1.48 1.28

0.90 0.70

0.36 0.32

3.15 3.77

256 260 285 303 409

17.80 13.72 1.33 7.49 0.66

27.73 35.88 2.02 16.78 1.98

75 96 340 300 1824

36 20 170 78 265

16.04 264.86 18.67 27.37 87.08

7.88 41.01 10.15 8.70 32.85

5.78 66.6 8.51 11.6 26.4

52.8 26.6 52.3 28.4 11.6

Mean ±

8.20 7.54

16.88 15.18

527 735

114 103

82.80 105.84

20.12 15.64

22.59 26.07

34.3 17.9

-1

-1

copies RNA·cell

day

%

AGMs

Mandrills

Macaques

41

4.4 Simulations The optimal fit for model parameters was accomplished by using nonlinear fitting software "Madonna" (Oster and Macey, 2000). The optimal fit of the viral burst-size (p/c) yielded higher values, 1,117 copies RNA·cell-1 on average, only 38% higher than values derived from analytical solutions. Values of the target cell death rate constant d obtained by nonlinear fitting were around 10-2-10-3 day-1 for all specimens, 10-100 times the values determined by kinetic analysis. Simulations using fitted values from table 4i overestimated the viral steady state by 1-2 log in African primates (see figures 4e and 4f). No significant change was seen after the optimal fit of target cell loss rate constant in Asian primates was derived. Numeric integration solutions using calculated parameter solutions were quite accurate. The only parameter that needed optimization In order to successfully recreate the data up to day 21, the optimal fit for the viral burst size (p/c) was generally larger than analytic estimations, 136±45% on average. This underestimation of p/c may stem from its dependence on absolute target cell numbers or caused by the effect of the quasi steady state assumption. After day ~21 the oscillatory behavior of the basic model becomes evident in the numeric integration simulations, and analytic predictions are no longer applicable. Therefore, this model is unable to accurately determine the target cell loss rate constant, which is a function of the viral steady state, and predicted by simulation to be 1-2 log higher than clinical data exhibited. Figures 4f to 4h depict the numerical integration simulations generated by "Madonna" (Oster and Macey, 2000) utilizing calculated values of δ and β, assuming a ten-fold decrease in k, and fitted parameter values for p/c and d in table 4i. During simulations, the viral clearance rate constant was arbitrarily assumed to be 6,000 copies RNA·ml-1·day-1, since values ranging between 3·103-1·104 copies RNA·ml-1·day-1 generated results not significantly different from one another.

42

Table 4i. Optimized values of viral burst size and target cell death rate Calculated values Optimized values -6 p/c d p/c d β (·10 ) δ day-1

µl·copies RNA-1·day-1

copies RNA·cell-1

day-1

copies RNA·cell-1

day-1

96001 96008 96011 96023 98007 98008 98011

0.51 1.57 1.42 0.99 0.66 0.58 0.76

7.93 6.56 2.10 20.28 31.89 40.04 3.31

261 729 1251 78 54 18 413

6.31·10-4 1.24·10-3 6.21·10-4 1.92·10-3 2.5·10-4 5.48·10-4 1.84·10-4

309 1057 1481 102 28 -382

5.45·10-3 2.07·10-2 1.81·10-2 1.79·10-2 7.27·10-3 ---

Mean ±

0.93 0.42

16.02 15.04

400 451

7.70·10-4 5.66·10-4

560 530

1.39·10-2 6.25·10-3

12A4 10G 12C2 2C2

1.62 0.68 0.92 1.25

1.70 18.24 7.72 0.90

2025 121 383 1052

1.92·10-4 1.33·10-3 6.68·10-4 6.19·10-5

1849 197 552 9471

1.95·10-2 1.98·10-2 1.24·10-2 1.85·10-2

Mean ±

1.12 0.41

8.43 7.01

895 849

5.64·10-4 5.75·10-4

3017 4361

1.76·10-2 3.48·10-3

256 260 285 303 409

0.24 0.14 0.22 0.29 0.27

26.52 10.01 1.94 10.68 1.80

39 75 182 128 309

7.28·10-3 1.97·10-2 8.9·10-3 6.38·10-3 3.04·10-2

57 80 313 313 567

6.98·10-3 5.48·10-2 4.42·10-3 3.26·10-3 2.01·10-3

Mean

0.23 0.06

10.19 10.07

147 106

1.45·10-2 1.04·10-2

266 208

1.43·10-2 2.27·10-2

AGMs

Mandrills

Macaques

±

43

AGM 96011

9

log[copies RNA/ml]

8 7 6 5 4 3 0

50

200

AGM 96023

8

log[copies RNA/ml]

100 150 Days postinfection

7 6 5 4 3 0

50

200

AGM 98008

8

log[copies RNA/ml]

100 150 Days postinfection

7 6 5 4 3 0

50

100 150 Days postinfection

200

Figure 4f. African green monkey datasets shown as dots. Numerical integration solution in black solid line using parameter values from table 4i.

44

MND 10G

log[copies RNA/ml]

8 7 6 5 4 3 0

50

200

MND 12C2

8

log[copies RNA/ml]

100 150 Days postinfection

7 6 5 4 3 0

50

100 150 Days postinfection

200

MND 2C2

log[copies RNA/ml]

8 7 6 5 4 3 0

50

100 150 Days postinfection

200

Figure 4g. Mandrill datasets shown as dots. Numerical integration solution in black solid line using parameter values from table 4i.

45

MAC 303

log[copies RNA/ml]

8 7 6 5 4 3 0

50

200

MAC 260

8

log[copies RNA/ml]

100 150 Days postinfection

7 6 5 4 3 0

50

100 150 Days postinfection

200

MAC 256

log[copies RNA/ml]

8 7 6 5 4 3 0

50

100 150 Days postinfection

200

Figure 4h. Macaque datasets shown as dots. Numerical integration solution in black solid line using parameter values from table 4i.

46

5. Discussion Viral kinetic characteristics of the African primates show consistency with classic primary infection. Viremia peaks of SIV/AGM infections are equivalent to those of pathogenic virus-host interactions; however, viral steady states were significantly lower than pathogenic strains of primate lentiviruses. Generally, viral kinetics showed equivalent behavior to nonpathogenic interactions of HIV with humans and to primates exposed to genetically altered strains of pathogenic SIV viruses. Attempts to simulate primary infection in SIV infected African monkeys using the basic model ran into the same difficulties of other studies. Because parameter values are constrained by biological limits, simulation of this model generates many oscillations. Stafford et al. (2000) tried to supplement the basic model with functions that simulate an immune response. Other workers have enhanced the model with a proliferation term (Perelson and Nelson, 1999), which is based on the premise that the main mechanism of generation of target cells is through cell proliferation. Their endeavors have in general increased the oscillatory behavior of the model. Preliminary attempts to answer this issue are under way but are beyond the scope of this thesis. Nowak et al. (1997) determined the infected cell loss rate constant in macaques range between 0.56 to 0.90 by potent antiviral treatment. Little et al. (1999) corroborated the applicability of the SIV/ macaque model to the human question. Likewise, Perelson et al. (1996) showed average values of δ for HIV infected humans 0.49±0.13 which are comparable to SIV/macaque infections. Stafford et al. (2000) ascertained, by nonlinear fitting, the infected cell loss rate constant in humans infected with HIV during primary infection which ranged from 0.17 to 0.80. We find values for infected cell loss rate to range from 0.46-6.82 (1.86±1.59) for African primates, and 0.07-1.60 (0.64±0.59) for Asian primates. Our results reveal values of δ that are significantly lower by 2-5 times (p<0.005), than chronic SIV infection in its natural host. This leads to an interesting conclusion that a larger number of infected cells are being eliminated in the African primates. Since the virus-host interactions between SIV and its host are in general nonpathogenic, this may be a suitable model of non-progressing HIV patients which do not exhibit AIDS symptoms.

47

Although δ is higher in SIV-infected African primates, it seems that immune activity is not higher than in Asian primates or HIV-infected humans (Diop et al., 2000). The basic viral infection model treats every parameter as a "black box" and cannot determine actual mechanisms of infected cell loss. Consequently, we propose three major routes that could cause this difference in δ: 1) higher virus-associated cell death, 2) augmentation of infected cell apoptosis, 3) genetic immunity to the virus or 4) more efficient immune response that is not observed using current experimental technology. It appears likely that there is a qualitative difference between virus-host interactions. Interactions that cause a very high rate of infected cell loss are generally fatal to the host, like

Variola and Filoviridae (Ebola and Marburg viruses) human infections. On the contrary, very slow rates induce chronic infections ending in pathogenesis, such as HIV and HCV. In the middle of the spectrum are viruses that cause chronic infections but have no fatal outcome during the host's natural lifetime, for example SIV infections of its natural host, non-progressing HIV cases and all of the Herpes virus family. From an epidemiological point of view, Ebola-like behavior is analogous to a brush fire which burns hot and fast but dies quickly. This is not a good strategy for viral survival since the death rate of hosts is faster than secondary infections; nor for the individual since he/she succumbs to the virus. HIV-like strategy is quite good for the virus because it is transmitted to a large number of new hosts during its lifetime. Moreover, if the epidemic is quenched in one part of the world it can emerge somewhere else, especially in the era of modern transportation. The most productive approach to viral expansion is the chronic/nonpathogenic relations where the virus remains quiescent during the entire life span of the host during which it can cause many secondary infections. In fact, seroprevalence of SIV in natural hosts (~45%, Ohta et al., 1989) is larger than HIV (~0.6%, estimated from UNAIDS publication, 2000) and HCV (~2%, Di Bisceglie 1998); and Ebola-human infections are almost nonexistent between outbreaks. To date, antiviral treatment of HIV strives to block secondary viral infection of target cells. This leads to an exponential decline of viremia, associated with δ, below current assay detection levels. The virus remains present in the host, albeit at a much lower concentration. Complications occur, such as viral mutant escape, where it is impossible to continue further treatment. Drugs of 48

this kind also have prohibitive price tags for long term regimen, especially in third world countries. Another way to combat AIDS induced pathogenesis may be to cause an increase in infected cell loss without further CD4 stimulation. The purpose is to cause the virus to recede to a lowerthan-pathogenic threshold, but keep immune responses at normal levels. In this way, the patient will carry the virus all his or her life, but will not evince any debilitating symptoms. Further experimental procedures coupled with more complicated mathematical models will pave the way to future treatment regimes. The purpose of this study was to isolate the parameter or parameters distinguishing pathogenic from non-pathogenic virus-host interactions. The differentiating parameter is decidedly infected cell loss rate constant, in this case. Indeed, the lowest estimations of δ in African green monkeys and mandrills are still significantly higher than in macaques and humans. Even so, studies defining viral parameters were carried out in different ways and the differences could stem from this inconsistency. Notwithstanding, the kinetic and clinical results show large differences between African and Asian primates; as aforesaid, SIV is non-pathogenic in its natural host. Comprehension of the control mechanisms of African primates may enable us to use this information to the benefit of mankind.

49

6. Reference 1. Akari H, Nam K, Mori K, Otani I, Shibata H, Adachi A, Terao K, Yoshikawa Y. 1999. Effects of SIVmac infection on peripheral blood CD4+/CD8+ T lymphocytes in cynomolgus macaques. Clin Immunol 91:321-329. 2. Coffin JM 1995. HIV population dynamics in vivo: implications for genetic variation, pathogenesis and therapy. Science 267:483-489. 3. Crotty PL, Staggs RA, Porter PT, Killeen AA and McGlennen RC. 1994. Quantitative analysis in molecular diagnostics. Hum Pathol 25:572-579 4. Daniel MD, Letvin NL, King NW, Kannagi M, Sehgal PK and Hunt RD. 1985. Isolation of T-cell tropic HTLV-III-like retrovirus from macaques. Science 228:1201-1204. 5. Desrosiers RC. 1990. The simian immunodeficiency viruses. Ann Rev Immunol 8:557578. 6. Di Bisceglie AM. 1998. Heptitis C. Lancet 351:351-355. 7. Duesberg PH. 1995. Is HIV the cause of AIDS? Lancet 346:1371-1372. 8. Diop OM, Gueye A, Dias-Tavares M, Kornfled C, Faye A, Ave P,Huerre M, Corbet S. Barre-Sinoussi F and Müller-Trutwin MC. 2000. High levels of viral replication during primary SIVagm infection are rapidly and strongly controlled in African Green Monkeys. J Virol 74:7538-7547. 9. Easterbrook P. 1999. Chemokine receptors and HIV disease. J HIV Therapy 4:98-100. 10. Elton CS and Nicholson M. 1942. The ten-year cycle in numbers of lynx in Canada. J Anim Ecol 191:215-244. 11. Freed EO, Myers DJ and Risser R. 1990. Characterization of the fusion domain of the human immunodeficiency virus type 1 envelope glycoprotein gp41. PNAS 87:4650-4654. 12. Fultz PN, McClure HM, Anderson DC, Swenson RB, Anand R and Srinivasan A. 1986. Isolation of a T-lymphotropic retrovirus from naturally infect sooty mangabey monkeys (Cercocebus atys). PNAS 83:5286-5290. 13. Gao F, Yue L, Robertson DL, Hill SC, Hui HX, Biggar RJ, Neequaye AE, Whelan TM, Ho DD, Shaw GM, Sharp PM and Hahn BH. 1994. Genetic diversity of human 50

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51

23. Lernould JM. 1988. A primate radiation: evolutionary biology of African guenons. Cambridge University Press, Cambridge. 24. Letvin N and King N. 1990. Immunologic and pathologic manifestation of the infection of rhesus monkeys with simian immunodeficiency virus of macaques. J Acquired Immune Defic Syndr 3:1023-1040. 25. Little SJ, McLean AR, Spina CA, Richman DD and Havlir DV. 1999. Viral dynamics of acute HIV-1 infection. J Exp Med 190:841-850. 26. Lotka AJ. 1925. Elements of physical biology. Williams and Willkins. 27. Malthus TM. 1798. An essay on the principle of population. [Penguin books, 1970]. 28. Marx PA, Li Y, Lerche NW, Sutjipto S, Gettie A, Yee JA, Brotman BH, Prince AM, Hanson A, Webster RG and Desrosiers RC. 1991. Isolation of a simian immunodeficiency virus related to human immunodeficiency virus type-2 from a West African pet sooty mangabey. J Virol 65:4480-4485. 29. McDougal JS, Nicholson JK, Cross GD, Cort SP, Kennedy MS and Mawle AC. 1986. Binding of the human retrovirus HTLV-III/LAV/ARV/HIV to the CD4 (T4) molecule: conformation dependence, epitope mapping, antibody inhibition, and potential for idiotypic mimicry. J Immunol 137:2937-2944. 30. Mellors JW, Rinaldo CR Jr, Gupta P, White RM, Todd JA and Kingsley LA. 1996. Prognosis in HIV-1 infection predicted by quantity of virus in plasma. Science 272:11671170. 31. Microsoft. 1997. Miocrosoft Excel 97 – Hebrew Edition. 32. Mittler JE, Antia R and Levin B. 1995. Population dynamics of HIV pathogenesis. Trends in Ecology and Evolution 10:224-227. 33. Mittler JE, Sulzer B, Neumann AU and Perelson AS. 1998. Influence of delayed viral production on viral dynamics in HIV-1 infected patients. Math. Biosc. 152:143-163. 34. Müller-Trutwin MC, Corbet S, Dais-Tavares M, Herve VM, Nerrienet E, GeorgesCourbot MC, Saurin W, Sonigo P and Barre-Sinoussi F. 1996. The evolutionary rate of non-pathogenic Simian Immunodeficiency Viruses (SIVagm) is in agreement with a rapid and continuous replication in vivo. J Virol 223:89-102. 52

35. Murray JD. 1993. Biomathematics. Volume 19. 2nd ed. Springer-Verlag. Berlin. 36. Nowak MA, Lloyd AL, Vasquez GM, Wiltrout TA, Wahl LM, Bischofberger N, Williams J, Kitner A, Fauci AS, Hirsch VM and Lifson JD. 1997. Viral dynamics of primary viremia and antiretroviral therapy in Simian Immunodeficiency Virus infection. J Virol 71:7518-7525. 37. Ohta Y, Masuda T, Tsujimoto H, Ishikawa K, Kodama T, Morikawa S, Nakai M, Honjo S, and Hayami M. 1988. Isolation of simian immunodeficiency virus from African green monkeys and seroepidemiological survey ofthe virus in various nonhuman primates. Int J Cancer 41:115–122. 38. Oster GF and Macey RI. 2000. Berkeley Madonna (a numerical integration and fitting program). Version 7.0.3 39. Pantaleo G, Graziosi C, Demarest JF, Butini L, Montroni M, Fox CH, Orenstein JM, Kotler DP and Fauci AS. 1993. HIV infection is active and progressive in lymphoid tissue during the clinically latent stage of disease. Nature 362:355- 362. 40. Perelson AS and Nelson PW. 1999. Mathematical Analysis of HIV-1 Dynamics in Vivo. SIAM Rev 41(1):3–44. 41. Perelson AS, Neumann AU, Markowitz M, Leonard JM and Ho DD. 1996. HIV-1 dynamics in vivo: virion clearance rate, infected cell life span and viral generation time. Science 271:1582-1586. 42. Perko L. 1991. Differential equations and dynamical systems. Springer-Verlag, New York. 43. Phillips AN. 1996. Reduction of HIV concentration during acute infection: independence from a specific immune response. Science 271:497-499. 44. Sachsenberg N, Perelson AS, Sabine Y, Schockmel GA, Leduc D, Hirschel B and Perrin L. 1998. Turnover of CD4+ and CD8+ T Lymphocytes in HIV-1 infection as measured by Ki-67 antigen. J Exp Med. 187:1295-1303. 45. Sarkar FH, Li YW, Ball DE and Crissman JD. 1992. Comparative method for detection of RNA-PCR-amplified signals. BioTechniques 12:22,24,26.

53

46. Sattentau QJ, Clapham PR, Weiss RA, Beverley PC, Montagnier L, Alhalabi MF, Gluckmann JC and Klatzmann D. 1988. The human and simian immunodeficiency viruses HIV-1, HIV-2 and SIV interact with similar epitopes on their cellular receptor, the CD4 molecule. AIDS 2:101-105. 47. Souaze F, Ntodou-Thome A, Tran CY, Rostene W and Forgez P. 1996. Quantitative RTPCr: limits and accuracy. BioTechniques 21:280-285. 48. Sullivan KM, Pope S, Gill P and Robertson JM. 1992. Automated DNA profiling in fluorescent labeling of PCR products. PCR Method Appl 2:34. 49. Temin HM and Bolognesi DP. 1993. Where has HIV been hiding? Nature 3672:292-293. 50. UNAIDS 2000. Report on the global HIV/AIDS epidemic: December 2000. Joint United Nations Programme on HIV/AIDS. 51. Volterra V. 1925. Variations and fluctuations of a number of individuals in animal species living together. Translated from Italian in: Chapman RN. 1931. Animal Ecology. McGraw-Hill, 409-448. 52. Dicesare J, Grossman B, Katz E, Picozza E, Ragusa R, Woudenberg T. 1993. A highsensitivity electrochemiluminescence-based detection system for automated PCR product quantitation.Biotechniques 15:152-157. 53. Wang AM, Doyle MV and Mark DF. 1989. Quantitation of mRNA by the polymerase chain reaction. PNAS 86:9717-9721. 54. Wei X, Ghosh SK, Taylor ME, Johnson VA, Emini EA, Deutsch P, Lifson JD, Bonhoeffer S, Nowak MA, Hahn BH, Saag MS and Shaw GM. 1995. Viral dynamics in human-immunodeficiency-virus type-1 infection. Nature 373:117-122. 55. Wiesner RJ. 1992. Direct quantification of picomolar concentrations of mRNA by mathematical analysis of a reverse transcription/exponential polymerase chain reaction assay. Nucleic Acids Res 20:5863-5864. 56. Zhang X and Neumann AU. 2001. The periodic solution of viral dynamics in vivo. In press. 57. Zhang ZQ, Schuler T, Zupancic M, Wietgrefe S, Staskus KA, Reimann KA, Reinhart TA, Rogan M, Cavert W, Miller CJ, Veazy RS, Notermans D, Little S, Danner SA, Richman 54

DD, Havlir D, Wong J, Jordan HL, Schacker TW, Racz P, Tenner-Racz K, Leetvin NL, Wolinsky S and Haase AT. 1999. Sexual transmission and propagation of SIV and HIV in resting and activated CD4+ T cells. Science 286:1353-1357.

55

‫תקציר‬ ‫הדבקת נגיפי‬

‫‪SIV‬‬

‫במאכסניהם הטבעיים‪ ,‬הקופים האפריקנים‪ ,‬אינה פתוגנית‪,‬‬

‫ואילו הדבקתם בפרימטים זרים כמו קופים אסיאתיים מובילה ל‪ ,AIDS-‬בדומה ל‪-‬‬ ‫‪HIV‬‬

‫בבני אדם‪ .‬המודלים המתמטיים‪ ,‬אשר מהווים כלים רבי עצמה במחקר‬

‫הדינמיקה הנגיפית‪ ,‬שינו את הבנתנו אודות מנגנונים נגיפיים וכוונונו לטיפול‬ ‫תרופתי אנטיוירלי הולם‪ .‬מחקר זה הוא הראשון העוסק באנליזת הדינמיקה‬ ‫הויראלית של‬

‫‪SIV‬‬

‫במאכסן הטבעי‪ ,‬קופים אפריקנים‪ .‬אינטרקציות וירוס‪-‬מאכסן‬

‫במהלך ההדבקה הראשונית‪ ,‬הקיימות בקופים אלה‪ ,‬נראו דומות לאשר התגלו‬ ‫בקופים אסיאתיים בתחום הוירולוגיה והקינטיקה‪ ,‬אך שונות בתחום האימונולוגיה‬ ‫והפתולוגיה‪.‬‬ ‫מעקב אחרי דינמיקת‬ ‫‪ (sabaeus‬במנדרילים‬

‫‪SIV‬‬

‫בקופים אפריקנים ירוקים )‬

‫)‪Mandrillus sphinx‬‬

‫‪Cercopithecus aethiops‬‬

‫‪ (Cercopithicidae‬ובמקאקים‬

‫)‪mulatta‬‬

‫‪(Macaca‬‬

‫נעשה במהלך הדבקה ראשונית עד ליום ‪ ,360 ,84‬ו‪ ,100-‬בהתאמה‪ .‬נעשה שימוש ב‪-‬‬ ‫‪,RT-PCR‬‬

‫‪ PCR‬ו‪FACS-‬‬

‫למדוד ריכוזים של‬

‫‪ RNA‬ו‪DNA-‬‬

‫ולימפוציטים מסוג‬

‫‪CD4‬‬

‫בפלסמת הדם ובקשרי הלימפה‪.‬‬ ‫עבודה זו משתמשת בקירובים המופקים מהמודל המתמטי והמבוססים על‬ ‫אנאליזה קינטית כדי לקבל ערכים מספריים עבור פרמטרים נגיפיים‪ .‬ההנחות‬ ‫והפתרונות האנליטיים אינם יחודיים‪ ,‬אך השימוש בהם בהקשר זה הוא אכן ייחודי‪.‬‬ ‫ממצאינו מראים שזמני ההכפלה הוירליים היו‬

‫)‪0.85(±0.27‬‬

‫ימים‪ ,‬בכל הקופים‪ .‬כן‪,‬‬

‫קצב הדבקת תאי המטרה היה באותו סדר גודל בכל הקופים‪.4.06·10-3-2.01·10-3 ,‬‬ ‫הערכות עבור מספר הויריונים הנוצרים מכל תא מודבק‪ ,‬שהראו שונּות גדולה‪ ,‬נעו‬ ‫בתחום ‪ ,10-3583‬ללא קשר למוצא הקוף‪ .‬ממוצע זמני מחצית‪-‬החיים של תאים‬ ‫מודבקים היה ‪ 1.86‬ימים )בטווח ‪ 6.82-0.46‬ימים( בקופים האפריקנים‪ ,‬ואילו המקאקים‬ ‫א‬

‫הראו ממוצע ערכים של‬ ‫‪reproductive numbers‬‬

‫‪0.64‬‬

‫ימים )בטווח‬

‫‪1.60-0.07‬‬

‫בקופים אסיאתיים היו גבוהים פי‬

‫‪2.5‬‬

‫ימים(‪ .‬בהתאמה‪,‬‬

‫‪basic‬‬

‫מאשר בקופים אפריקנים‪.‬‬

‫כמו כן‪ ,‬היחס בין התאים המודבקים בתוך אוכלוסיית התאים הכללים היה גדול פי‬ ‫מאה בקופים האסיאתיים‪.‬‬ ‫מעניין‪ ,‬שההערכות עבור קצב איבוד תאים מודבקים בקופים אפריקנים גדולות‬ ‫פי‬

‫‪3‬‬

‫כמעט‪ ,‬בממוצע‪ ,‬מאשר בקופים אסיאתיים‪ ,‬בעוד שפרמטרים אחרים לא היו‬

‫שונים באופן משמעותי בין שני המודלים האנימליים האלה‪ .‬המסקנה העולה מנתון‬ ‫זה היא שהתוצאה הבלתי‪-‬פתוגנית בהדבקת‬

‫‪SIV‬‬

‫במאכסן הטבעי שלו קשורה‬

‫באיבוד התאים המודבקים המייצרים וירוסים‪ .‬אנו מציעים הבדל איכותי בין‬ ‫הדבקות נגיפיות כרוניות‪ ,‬אשר הפתוגניּות של אינטרקציות וירוס‪-‬מאכסן מוכתבת‬ ‫על ידי פרמטר זה‪ .‬מנגנונים אנטיוירליים פוטנציאלים הנתונים למחקר הם תפקוד‬ ‫לקוי תאי‪ ,‬אפופטוסיס‪ ,‬גנטיקה ותגובות חיסוניות תאיות‪ .‬ממצאים אלה חשובים‬ ‫בהבנת יחסי הנגיף‪-‬מאכסן בחולי‬

‫‪HIV‬‬

‫שאינם מתקדמים ל‪ ,AIDS-‬ולהנחות המשך‬

‫המחקר בתרופות אנטי‪-‬ויראליות‪.‬‬

‫ב‬

‫תוכן עניינים‬ ‫תקציר ‪ .....................................................................................................‬א‬ ‫‪ .1‬מבוא ‪1 ...................................................................................................‬‬ ‫‪ 1.1‬נגיף הכשל החיסוני הסימיאני )‪3 .............................................. (SIV‬‬ ‫‪ 1.1.1‬מבנה הוריון ‪5 .......................................................................‬‬ ‫‪1.1.2‬‬

‫מעגל החיים הוירלי ‪7 ...........................................................‬‬

‫‪ 1.1.3‬חדירה אל התא ‪7 .................................................................‬‬ ‫‪ 1.1.4‬סינטזת הפרו‪-‬וירוס ‪8 ............................................................‬‬ ‫‪ 1.1.5‬מעבר אל הגרעין‪ ,‬אינטגרציה וביטוי גנים ‪9 ............................‬‬ ‫‪ 1.1.6‬הרכבת הוירוס ‪10 .................................................................‬‬ ‫‪ 1.2‬קופים אסיאתיים ‪10 ........................................................................‬‬ ‫‪ 1.3‬קופים אפריקנים ‪11 ........................................................................‬‬ ‫‪ 1.4‬מודלים מטמתיים ‪14 .......................................................................‬‬ ‫‪ 2‬חשיבות המחקר ‪16 .................................................................................‬‬ ‫‪ .3‬שיטות ‪16 ...............................................................................................‬‬ ‫‪ 3.1‬מדידות של ‪ RNA‬ו‪ DNA-‬נגיפיים ולימפוציטים מסוג ‪16 ............... CD4‬‬ ‫‪ 3.1.1‬מדידות ו‪16 ................................................................. DNA-‬‬ ‫‪19 ...................................... Real Time Polymerase Chain Reaction 3.1.2‬‬ ‫‪19 .................... Fluorescence-Activated Cell Sorter analysis (FACS) 3.1.3‬‬ ‫‪ 3.2‬קבוצות של קופים ‪21 ......................................................................‬‬ ‫‪ 3.2.1‬קופים אפריקנים ירוקים ‪21 ....................................................‬‬ ‫‪ 3.2.2‬מנדרילים ‪21 .........................................................................‬‬ ‫‪ 3.2.3‬מקאקים ‪21 ..........................................................................‬‬ ‫‪ 3.3‬המודל ‪21 .......................................................................................‬‬ ‫‪ 3.3.1‬הצגת מישור הפזה ‪24 ..........................................................‬‬ ‫‪ 3.3.2‬מצבים נייחים‪24 ...................................................................‬‬ ‫‪ 3.3.3‬יציבות המודל ‪25 ..................................................................‬‬

‫‪ 3.4‬אינטגרציה נומרית ‪26 .....................................................................‬‬ ‫‪ 3.5‬התאמה ‪27 .....................................................................................‬‬ ‫‪ .4‬תוצאות ‪28 ..............................................................................................‬‬ ‫‪ 4.1‬פתרונות אנאליטיים ‪28 ...................................................................‬‬ ‫‪ 4.1.1‬שיפועים והערכת פרמטרים ‪28 ..............................................‬‬ ‫‪29 ............................................... R0 - Basic Reproductive Ratio 4.1.2‬‬ ‫‪ 4.1.3‬הערכה עבור ‪29 ............................................................. Imax‬‬ ‫‪ 4.1.4‬הערכות עבור פרמטרים נוספים ‪29 .......................................‬‬ ‫‪ 4.2‬אנאליזה קינטית ‪31 ........................................................................‬‬ ‫‪ 4.2.1‬נתונים אודות קשרי לימפה ‪32 ...............................................‬‬ ‫‪ 4.2.2‬קופים אפריקנים ירוקים ‪32 ....................................................‬‬ ‫‪ 4.2.3‬מנדרילים ‪32 .........................................................................‬‬ ‫‪ 4.2.4‬מקאקים ‪33 ..........................................................................‬‬ ‫‪ 4.2.5‬גרפים וטבלאות ‪33 ...............................................................‬‬ ‫‪ 4.3‬הערכות עבור פרמטרים דינמיים ‪37 ................................................‬‬ ‫‪ 4.3.1‬זמן הפלה וירלית )‪37 ..................................................... (1/r0‬‬ ‫‪ 4.3.2‬קבוע קצב איבוד תאים מודבקים )‪37 .................................. (δ‬‬ ‫‪ 4.3.3‬קבוע קצב הדבקת תאי מטרה )'‪37 .................................... (β‬‬ ‫‪ 4.3.4‬פרמטרים נוספים ‪37 .............................................................‬‬ ‫‪ 4.3.5‬טבלאות ‪38 ...........................................................................‬‬ ‫‪ 4.4‬הדמיות ‪42 .....................................................................................‬‬ ‫‪ 5.‬דיון ‪47 ....................................................................................................‬‬ ‫‪ 6.‬מקורות ‪50 ..............................................................................................‬‬

‫אוניברסיטת בר‪-‬אילן‬

‫דינמיקה של נגיף הכשל החיסוני הסימיאני )‪(SIV‬‬ ‫במהלך הדבקה ראשונית בקופים אפריקנים‬

‫דוד בר‪-‬גולני‬

‫מוגש כחלק מהדרישות לקבלת תואר מוסמך )‪ (M.Sc.‬בפקולטה‬ ‫למדעי החיים אוניברסיטת בר‪-‬אילן‬

‫רמת גן‪ ,‬ישראל‬

‫‪2001‬‬